Yes you can, but you want to make sure you use the right type of paint and probably a roller instead of a brush. When you paint laminate the brush strokes show up much more than they would on wood. I'd say an epoxy paint would be best from my own experience just so that it really sticks. Especially when painting a bathroom that will have to stand up to water and steam you want to make sure that paint won't peel right off!

I agree, you can paint the cabinet, we use a melamine paint, somewhat difficult to use due to it's make up. It's applied with a foam roller in one direction, you can't over roll the paint. I suggest doing this in a garage or outside due to the higher toxicity than normal paints.
Good luck

You can definitely do it but I'd suggest to use a paint gun to have a better quality finish, regardless of what type of paint you end up using. If you don't have one, you can rent the compressor from a big box store. It will pay back at the end. Good luck,
Laura

Scuff sand the laminate, prime with oil based, low odor Kilz. Topcoat with Sherwin Williams oil based low odor Pro-Classic.