Not all false friends are "false cognates". Many of those which cause problems between English and other European languages are real cognates - words in different languages with similar forms sharing a common etymology - where one has undergone a semantic shift and the other has not. For example, the English adjective "actual", meaning "real", in contrast with imaginary or predicted, has true cognates in most Western European languages - e.g. French actuel, German aktuel, Italian attuale, Dutch actueel - all of which mean "current", of the moment. In this case it was the English word which changed its meaning in or around the eighteenth century. Eventually (after a long delay) and its cognates eventuellement, eventualmente, eventueel etc. (possibly, maybe) are another common example along the same lines.
Actual false cognates - homonyms or near-homonyms in different languages with distinct meanings and unrelated etymologies - are somewhat rarer, at least in fairly closely related languages like those of Europe (apart from a few non-indo-European isolates). Examples might be the (unlikely) confusion between butter and donkeys in Italian and Spanish (es burro = it asino; it burro = es mantequilla), or an English kiss and a Finnish cat, or French and English meanings of chair.