Can beer be tainted with nethanol

I have a phobia of consuming methanol tainted beer because very often the same bars sell tainted cocktails and other high alcohol drinks.

I have had beer that tasted weird in the past and also beer with foam that would remain in the bottle and not resolve.

But is it even possible to taint it with methanol and not mess up the taste completely? Perhaps by using a percentage of very cheap beer + water + methanol.