For Borderlands 1, maximum bonus was +4/3/3, making maximum level for single skill 9. For Borderlands 2 there is no specific information on any of the wikis, except for "Legendary" and "Slayer of Terramorphous" mods, which give +5 and +4 respectively.
What are maximum skill bonuses from any Class Mods?
Best Answer
Some non-legendaries have +6 to a skill. Never seen more than 6, though from my inventory even a green can have a level 6, so I don't think it's tied necessarily to rarity (it may be to level though, these are level 50 mods and I think lower level ones go as low as +1). I've played 4 full playthroughs (two playthrough ones, two playthrough twos) and I don't recall ever seeing a single +7, though I have a few +6s in my inventory right now.
The skill counts are the same as the first game; white mods have no skills they boost, greens boost 1, blue 2, purple 3.
According to The Gear Calculator (via Orc JMR), green Mods can grant up to +5, blue Mods up to +6/5, purple Mods up to +5/4/4. So Purple mods boast the most skill points overall, but blue mods can actually boost two specific skills higher than a purple can.
Legendary mods have set skills they boost, the Legendary (Class) mod boosts by +5, the Slayer of Terra mods have +4.