I was working on the assumption that it didn't matter where the bonus came from, only the final value was really important.
In the model I would adopt, attribute mods would be capped at +3, and Fighters would get a +2 to all attacks, giving the +5.
Additionally, they would get a +1 per 2 levels, as in 4e, and would also get their first feat at 2nd level, which could give them a further +2 at that point.
I could, potentially, see characters getting a masterwork weapon or a racial modifier giving a further +1 here or there... Mostly, though, I'd prefer any bonuses higher than that to be situational - a "precise attack" power, or a buff spell, or flanking, or whatever.
See, I disagree. On a d20 system, there's a sweet spot for modifiers at about +10. That's the point where your total result is roughly equally split between 'luck' and 'skill'. (Mathematically, it makes no difference - d20+10 vs DC 20 is the same as d20+50 vs DC 60, but the former feels better.)
The problem is that if you allow your 1st level character to hit that sweet spot immediately (and on all attacks, without any sort of "clever play bonus"), there really isn't anywhere for the character to grow to. Far better to start the character at a point where they are reasonably competent but only showing potential (rather than being the finished article), then fairly quickly grow them to the sweet spot.
The other consideration is the role of specialisation in the system. Both 3e and 4e gave players a huge range of options for advancing their characters, and I wouldn't bet against 5e eventually offering the same. But with all those options, as the game progresses the gap between the specialist and the non-specialist will grow. And if it grows too large, it can become a real issue, ultimately breaking the math in the game (as Andy Collins discussed at length in the "Epic Level Handbook" - a book that was widely slated, but which has the core advancement mechanic that was later adopted in 4e).
Far better to limit specialisation at the outset, and allow characters to grow reasonably specialised as they go. Unless you were thinking of having +10 for the ultra-specialist, +8 for the 'common' Fighter, +6 for 'everyone else', and +4 for the weakling mage. But then, I see absolutely no benefit in not reducing all those numbers by at least 4 across the board, and since people do better adding smaller numbers than bigger ones, I can see at least one advantage.
The other advantage of not allowing excessive specialisation at the outset is that if the campaign goes in a different direction from that expected, the non-specialist can adapt; the specialist is stuck. That's not good for anyone - either the player now has to not play the character he wanted, or the player has to play a character that doesn't suit the campaign, or the DM and other players are constrained to not take the game in that unexpected (and potentially interesting) direction.
