It looks like the sample rate is more than high enough to differentiate between the typical alpha, theta, and delta waves, which should be enough to discern sleep stages. The hard part is a quality electrode-scalp contact which maintains relatively constant impedance over the course of sleep, especially after movement. EEG is all about signal/noise, because ultimately you are amplifying tiny currents which are easily drowned out by noise from contact movement and I think even RF interference (you can trivially pick up 60hz mains noise even if you're sitting ≈10 feet from the nearest powered appliance/wire).

OpenBCI sells electrode caps which they claim are sufficient for sleep studies, unfortunately IIRC they are somewhat inconvenient because the electrodes require wetting and/or gel which must be applied to each electrode every time the cap is worn. I have yet to shell out the $500+ to give it a try, but that's still way cheaper than medical grade devices.