Yes, a bridge can certainly be removed. However, are you really asking whether it can be removed or can it be removed and then recemented onto the teeth?
If the dentist is going to remove the bridge to replace it with another, then it is usually drilled off. However, if he is trying to take it off to check or change something, and then wanting to recement it, then the technique is to try to tap it off with an instrument. Although that technique is sometimes successful, it also runs the risks of injuring the underlying tooth and even fracturing it off. I do not use that technique in my office because of the high risk of breakage.
However, there is another approach that has a lower rate of potential damage. The dentist will have you bite down on a warmed soft wax-like wafer, then cool it down with water. When it is cooled down, it is hard, and sticks to both the bridge and opposite teeth. Then the dentist will ask the patient to open his/her mouth quickly. Sometimes the bridge will come off, but on occasions it can loosen other crowns off of their underlying teeth. Clearly a better way to go than trying to tap off the bridge, but safest approach is for the dentist to just drill off the bridge and make you another one.