Good preparation is the key to success. If you take your time to prepare your surface before slapping on the paint, then it will last for years.
REMOVE the wallpaper!!! This can be done either by chemical application or a steamer & scraper. Ensure that the area to be painted is free of wallpaper bits and dust. You can also wash the surface with 'sugar-soap' after removing the wallpaper to ensure the resin/glue is off.
Depending on the surface type, you might have to apply a primer 1st before painting!