Valerie...I always remove wallpaper before I paint
Yes, you can. I would not recommend it though. When you paint over the wallpaper, the moisture from the paint may make the wallpaper bubble and become uneven, not to mention you will likely see the seams in the paper more than you did prior to painting. If later you or someone else want to remove the wallpaper, it will be more difficult than if you'd not painted over it. It would be best to remove the paper, insure all the remaining glue is cleaned off and then paint.
You can, but it will eventually look crappy due the paper lifting. Walmart and others sell a tool, PaperTiger, that is super for removing wallpaper. It is a scoring tool that will make perforations thru the paper so that hot water or paper removal solution can penetrate all underneath the paper enabling it to be pulled off relatively easy. Run the tool over the walls, squirt bottle to spray your solution on after scoring, and a little patience will get the job done nicely. Be sure to wipe up the water or solution as you go.
Just because you can, does not mean you should. Remove it . . . The end result will look far more professional. And if you paint over wallpaper and then have to take it down, and it will have to come off eventually, it will be 100 times harder to do so. Do yourself a favor and do it right the first time, then be proud of your work!
I painted a vinyl wallpaper that was in a bathroom. I worked very well and has been on there for about 10 years now. However, I would not do it to regular wallpaper.