You sure can paint over stubborn wallpaper! Use a fine grit sand paper to smooth any edges from bits you have been able to remove, then use a coat of a good quality primer to keep any pattern colour or bold lines from bleeding, then paint away-use a roller for a smooth application, or a brush if the paper is textured.
Some wall papers are very hard to remove. Usually scoring the paper will let the walll paper remover in behind the paper to do the job. As for painting I would ask the paint folks at a home center or paint company for their suggestions before I made too much of a mess.
About 40-50 years ago when I was a really young kid, people just plain painted over wall paper. Of course back then the paint was all oil base, and I don't know if that would make any difference or not.
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