Yes a Pokémon with a Choice Item can increase it's Stats but you can't increase a Stat more than 6 Stages.
So for example if you had a Pokémon with Belly Drum and Baton Pass, and you Baton Passed Belly Drum (which maximises your Attack, raising it by 6 Stages) onto a Pokémon with a Choice Band it will have + 6 in it's Attack Stat not + 7.
Why? Because Stats cannot be increased any further than + 6 Stages, the same as a Stat cannot be decreased by more than - 6 Stages.
If a Pokémon tries to increase one of it's Stats by more than 6 Stages, the move will fail. So if a Lucario uses Swords Dance three times to get a + 6 increase in it's Attack Stat, if it uses Swords Dance one more time the Move will fail unless Lucario has been switched out.
Also note that if you did increase your Stats while holding a Choice Item you would be locked into it and wouldn't be able to Attack so you would have to switch out. That's why I gave an example of Baton Passing Stat Boosts onto a Pokémon with a Choice Item.
Source: Experience and Here.
Hope I helped :)