It's a good idea to have two or three bottles, I sometimes have two bottles in the cage at the same time, just in case one stops working (the little ball bearing inside can get stuck after some time or sometimes one has emptied itself after our hamster has knocked it and unscrewed it partly!)
I'm a bit paranoid about water bottles letting enough out as well. One trick is to not completely fill the bottle - only fill it about 2/3 full so there's a gap at the top - apparently that's supposed to help the water come out easier.
Then as Cypher says, tap the spout when it's in the cage in position and check that water is coming out ok. Leaving the bottle very slightly unscrewed can also increase the water flow a bit but I find that's a bit of a fine line - too much unscrewed and it drips or empties itself. Not enough and the water doesn't come out so well. Bottles vary as well as to how well they work. I use these ones now and find they work better than many others I've tried - they don't drip and they always let water out, but even with these I find after a few months they don't work as well as when they're new. I buy two at a time to save on postage. They are also easy to get in and out as they come with a clip that you attach inside the bars, and then the bottle just clips in and out of the clip easily so quick to take out and put back.
Fpi 4660 Drinky75 Drinky B.
This is the only place I've found that sells the 75ml small mouse one (it's a good size for a hamster). Amazon sometimes sell them but only by marketplace sellers and often at ridiculous prices. They're quite cheap so getting a couple at a time can be good.
With these I find you can have them fully screwed up and they still let water out well if you only fill them 2/3 full.