What we call the Romance languages are a continuation of vulgar Latin. It's probably more accurate to say that vulgar Latin always had regional dialects, and that during late antiquity and the early middle ages those dialects started to splinter and became even less mutually intelligible than they had previously been. Part of the problem with studying this splintering is that this period of European history was poorly recorded.

During the period between the fall of the Roman Empire and the rise of modern nation states, their was probably a large dialect continuum across much of Western Europe. People could understand people from nearby villages and towns, but people from further away villages and towns would have been more difficult to understand (although someone from the middle might have been able to understand people on both sides).

Eventually certain local vernaculars became more important, usually because of a combination of state support, cultural influence, and trade, and these vernaculars also developed a written form (for vulgar Latin was almost always spoken). For example, Portuguese was the vulgar Latin dialect of the Northwestern Iberian peninsula, which was somewhat influenced by Arabic domination (and influenced by earlier Germanic conquerors). After this period of isolation, outside influence, and independent development, Portuguese came to be considered a language.

To answer your question, absolutely. The Catholic Church is of gigantic cultural importance in Western Europe, and it used Latin at mass until the 20th century. Even setting that example aside, it's not like people ever suddenly started speaking differently. Some people probably spoke in a dialect that you could argue to be either a late form of vulgar Latin or an early form of a Romance language. Documents often used written Latin alongside written forms of the new vernacular dialects. For example, the Oaths of Strasbourg (made by two Frankish kings) were written in both Latin and Old French (as well as Old High German).

Education was the major area outside of the Church that used Latin after vernacular languages had taken over much of daily life. The Latin Quarter in Paris obtained its name from the number of colleges/universities that all used Latin.

I have also heard that American universities would use Latin. David Levy mentioned it in his history of the University of Oklahoma, for example. The university had basic requirements that involved students having experience. I have heard that some universities actually used Latin in the U.S. but since I have not read much into it I won't make a claim that it was rare or common.

Education in Italy still has some schools that stress Latin and Greek in a Classical/Liberal Arts education. I think a school that does is called a Liceo Classico.