When you say "pressure", I think you are probably referring to something like "the height to which the water shoots when you operate the fountain". There are actually two things that affect this:
- The pressure of the water supply.
- Any obstruction to the pipe (such as corrosion).
As a thought experiment, imagine that you have a completely clear pipe except for an almost complete obstruction ten feet upstream of the fountain. When no water is flowing, the pressure is the same upstream and downstream of the obstruction. When you press the button, water will shoot out of the fountain at that pressure, but the water won't be immediately replaced, because the obstruction slows the flow. So the pressure on your side of the obstruction will drop, and the flow will slow to a trickle. When you release the button, water will continue to flow (slowly) past the obstruction until the upstream and downstream pressures are identical again. (How long the spurt lasts depends, among other things, on how springy the material pipe is: the more it stretches under pressure, the more water it will be able to store on your side of the obstruction).
Of course in real life whatever is obstructing your pipe will be less extreme, and distributed more along the length of the pipe. All the same, you will be able to distinguish between low pressure or an obstructed pipe by seeing to what extent you are getting "spurt, then slow" behaviour.
If your drinking fountain shoots water up into your eye but almost immediately drops to a trickle, an obstruction is the cause.
An indication of low pressure of the water supply as the cause is that the pressure (and so the flow) is different at different times. This depends where your water comes from. If there are pumps directly in the supply then they will pump until a desired pressure is reached, then stop until the pressure drops. That will give you some fluctuation. If you are being fed directly from a tank on the roof (cistern plus ball valve) then the pressure will vary hardly at all. If you are being fed from a distant tank shared with a lot of other water users, the pressure will depend on how many other people have their taps on at the same time as you. You say you have eliminated the last possibility in your case, but I include it here for the benefit of others in similar situations.