In reply to ryanrawlings: Belay devices work as friction multipliers. When there is a large angle between the live rope and the dead rope (i.e. the classic locked off position) a small force on the dead rope can hold a large force on the live rope. When there is a smaller angle, the ratio of forces between the dead and live ropes reduces until when both ropes are parallel the belay device is essentially acting as a pulley.
When you belay you continually alternate from the locked off position (close to 180 degrees between dead & live ropes) to a feeding position (smaller angle between dead & live ropes) where you can take in or feed out rope. That feeding position varies slightly between belay devices.
For example, with an old fashioned Sticht plate you need to have less than 30 degrees between the dead and live ropes in order to feed any rope. Conversely with an ATC you may only need to reduce the angle to close to 90 degrees before you can start to feed. It also depends considerably on the rope but most modern devices are fairly 'slick' so you don't need to reduce the angle massively.
In practice what varies therefore is how high you need to lift the hand on the dead rope relative to the belay device before you start to pull rope through and then how high the trajectory of that hand then needs to be. If a device is sticking you are possibly not raising your hand quite far enough to reduce the angle and hence the friction.
However, there are half a dozen other things that people can do whist belaying that can result in it not working as smoothly as it should, so that may not necessarily be the problem.
Anyway, hope that make some sense.