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Some people like bounded accuracy and some people don't.  Reading through the forums, the criticisms against bounded accuracy seem to fall into 2 categories:

1.  Robin Hood Scenario - a novice is as good at making the shot as an expert
2.  ACs are too low for powerful monsters - level 1 characters can easily hit Asmodeous

The proponents of bounded accuracy complain that high level characters are invincible since their AC is so high they can't be hit.

I think there are ways to improve bounded accuracy and satisfy both groups.  If we accept that some form of bounded accuracy is going to stay, here are some possible improvements:

1 - Get rid of the "10" as the standard, starting AC.  D&D has always assumed that an average unarmed and unarmored person had an AC 10.  In a bounded accuracy system, a minimum of 10 really collapses the useable real-estate if you stick with the D20 model.  Make the standard, starting AC a 2 so that only a 1 fails.  This may sound crazy at first but think about it.  For non-ranged attacks, it's actually really easy in real life to hit someone who is "standing and fighting" and not dramatically moving away or flat out running away.  Now, just because you hit someone doesn't mean that you actually did any serious damage.  So a low AC 2 (95% chance to hit) isn't really as ridiculous as it sounds. Whether or not you do any real damage could easily be handled on the HP side of the house (which is addressed below).  If the minimum AC is 2, there are several other things that must also take place to make this effective.

2 - Dramatically lower the AC bonus of armor.  They should look something like:

Padded armor = 3 + Dex modifier
Leather armor = 4 + Dex modifier
Hide armor =  5 + Dex modifier
Studded leather = 6 + Dex modifier
Scale mail = 7 + Dex modifier
Ring mail = 8
Chain mail = 9
Splint = 10
Banded = 11
Plate = 12
Shield = +1 AC

3 - Add back in AC scaling with level.  This helps alleviate some of the issues with bounded accuracy.  A level 1 fighter in plate with a shield still only has an AC of 13 (a 40% chance of getting hit w/o bonuses).  A level 20 fighter in the same plate with the same shield has an AC of 23 (a 5% chance of getting hit w/o bonuses).  This way characters still get the benefit from experience without breaking the bank.  The level 1 fighter is easier to hit and the level 20 knight is much harder to hit, but not impossible to hit.

4 - Get rid of almost all bonuses to hit except from class and level progression.

4.A. First, get rid of the STR bonus to hit.  This isn't as bad as it sounds since bounded accuracy makes up for easy to hit ACs by adding more hit points.  So while a strong character will not hit more often than a weaker one, they will do more damage.  If all things else are equal, the stronger character still has a better chance of taking out their weaker twin.

4.B. Don't have Weapon proficiencies add bonuses;  have them prevent penalties.  If you are not proficient with a weapon and attempt to use it then you could suffer something like a -5 or -2 to your attack roll.  If you are proficient with the weapon you are using then your unmodified D20 roll is your attack roll.

4.C. Add back in the attack bonus scaling by level based on class.  This would help alleviate the Robin Hood syndrome and actually separate the classes from each other.  I imagine something like "martial characters get a 1/2 per level bonus to attack with weapon attacks. Arcane characters get a 1/2 per level bonus to attack with arcane attacks. etc."  This means at level 1 a fighter and wizard both have a 50% chance of hitting a target with an AC 10.  However, at level 20 a fighter has a 95% chance of hitting that same target and a 50% chance of hitting an AC 20 target with a weapon attack.  If a level 20 wizard tries to use a weapon (not their thing), they still only have a 50% chance of hitting the AC 10 target and only a 5% chance of hitting the AC 20 target.  However, if the wizard casts a spell then they would also have the 95% chance to hit the AC 10 target and 50% chance to hit the AC 20 target.

5.  Increase the ACs of tougher monsters.  If you have a scaling attack bonus, it would be OK to have the highest level monsters with ACs of 20 - 25.  A 20th character would have a 50% chance of hitting a creature with an AC of 20 and a 25% chance of hitting a creature with an AC of 25, while a level 1 character would only have a 5%  chance of hitting something like a demi-god or the Duke of Hell.  That doesn't sound unreasonable.

6.  Increase starting Hit Points.  If you are drastically decreasing ACs then it only makes sense to increase the PCs' starting HP unless you want a truly gritty campaign.  This goes back to the point of above that just because you hit someone doesn't mean it was significant. It could have been mostly blocked, only a glancing blow or simple scratch or mainly absorbed by the armor, etc.

7.  Add in mob tactics.  This mainly for NPCs and monsters, but it would address the issue of the level 20 character who doesn't fear low level monsters because they can never be hit.  Essentially, if the DM decides that NPCs or monsters form into a mob they can collectively attack a single target.  This would represent a character surrounded by goblins or a group of peasant archers shooting at a dragon.  It would speed up combat since you could roll one attack for the whole mob and still keep their attacks meaningful.  A mob rule could look something like "The mob makes a single basic attack against a single target with an additional +1 to its attack roll for every 5 creatures in the mob.  If the mob successfully hits its target, the target suffers normal damage from the single basic attack with an additional +1 damage for every 5 creatures in the mob.  If the mob misses, then the mob only inflicts half damage.  To be part of the mob, a creature must be able to attack the target in the same round as the rest of the creatures in the mob."

8.  Add in decaying ACs.  Essentially, each time a target is attacked in a round, they suffer a -1 to their AC. This addresses situations where the target is being ganged up on by multiple attackers who shouldn't or couldn't be categorized into a mob (like the PCs themselves).  This would be in addition to any bonuses gained from flanking, etc.  So, no matter how tough you are, the more you are attacked, the more vulnerable you are.  This prevents the level 20 character from totally wading through a group of level 10 drow mercenaries because they can't even be hit.

So, besides totally abandoning bounded accuracy, what other improvements could be made?

1.  Robin Hood Scenario - a novice is as good at making the shot as an expert

This isn't actually an issue; all melee capable classes gain a bonus to hit as they level up.  Are you perhaps thinking of the reverse, in that it's as easy to hit a high level target as a low level target? 

First, get rid of the STR bonus to hit.  This isn't as bad as it sounds since bounded accuracy makes up for easy to hit ACs by adding more hit points.  So while a strong character will not hit more often than a weaker one, they will do more damage.  If all things else are equal, the stronger character still has a better chance of taking out their weaker twin.

This would be a fine system assuming they actually make the ability score damage matter across all levels; right now it's a very small portion of damage at high levels.

8.  Add in decaying ACs.  Essentially, each time a target is attacked in a round, they suffer a -1 to their AC. This addresses situations where the target is being ganged up on by multiple attackers who shouldn't or couldn't be categorized into a mob (like the PCs themselves).  This would be in addition to any bonuses gained from flanking, etc.  So, no matter how tough you are, the more you are attacked, the more vulnerable you are.  This prevents the level 20 character from totally wading through a group of level 10 drow mercenaries because they can't even be hit.

This sounds like a nightmare to keep track of. 

+1 from me also on what wrecan said.

Also...

I love suggestions 1 and 2. Would like to see how 1 plays out especially. It makes a lot of sense that unless you have some terrible luck you can hit an unarmored human standing still more than half the time! Can't say enough how much I like this idea.