This also depends on your goals and where you are in training. If you're still relatively new, you can do a lot without gaining weight or even while losing it.
Once you plateau, it gets harder, but changing approach to training can still lead to improvements without the need to eat that much more than maintenance.
A beginner or someone on juice can certainly get stronger while cutting. Assuming it to be an intermediate lifter, it is highly unlikely he can get stronger while on a deficit if the same training regimen is used. However, introduce different variables/adaptogens (i.e. new routine), it is plausible to drop body fat while gaining strength.