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Did the Norse ever encounter the Romans?

No, because (like you said) "the Norse" did not really exist during Roman times.

The contemporary inhabitants of Scandinavia did not yet constitute a clearly distinguishable group within the greater Germanic cultural field, with the exception of the Gothic/East Germanic tribes, for most of the Roman period.

In terms of language, the consensus within modern Germanic philology is that the North Germanic and West Germanic languages had a common phase called Late Germanic, which lasted roughly until the second half of the 4th century CE. It's assumed that the main catalyst in the splitt between West and North Germanic occurred when tribes from Southern Sweden or Zealand/Funen settled Jutland; which is generally assumed to have been more oriented towards West Germanic prior to the Migration Period.

The material cultures in both Scandinavia and Northern Germany, Northwestern Poland and the Netherlands all belonged to the (Pre-)Roman Iron Age, which did not really end until the beginning of the 5th century CE. The culture of Scandinavia would have been more focused on the sea, when compared to that of the tribes that came to inhabit what is now Germany and would have had somewhat diminished acces to Roman trade due to being further away from the Roman limes, but would have otherwise remained very similar.

The question whether direct Roman contacts existed with the inhabitants of Scandinavia, is difficult to answer, but seems unlikely. The Romans (specifically Jordanes, writing in the 6th century) seemed to think that Scandinavia was an island; then again some historians think it is plausible that the Greek explorer Pytheas of Massalia reached the Baltic; which would certainly make it possible for the Romans to do the same several centuries later.

Regardless, a trade network linking Scandinavia to the Roman Empire all the way down to the Mediterranean certainly existed and many Roman artifacts have been found in Sweden and Norway.