Not really, if you define "good" as fluent. There are 24 "official" languages within the EU alone, plus many unofficial languages too. Some countries are better at teaching second and third languages than others, for example the English in particular are shockingly bad at teaching second languages well.
Most people will study English if that's not their native language and almost as many will study another of the major European languages throughout secondary/ high school. With some effort they can get by in that language later on. There are also generational divides too. English is now the preferred second language everywhere. The generation before mine it was German in northwestern Europe, and going further back that was true of large parts of non-German speaking central Europe too. For my own generation Russian was the most common second language of Eastern Europe, but now it's English.
Quite a few countries have more than one official language and usually there's a requirement to teach a basic knowledge of the of the "other" official language(s) if they're widely spoken within the country. A lot of people end up with a fairly good "passive" knowledge of the other language, ie, they can understand what's being said and read a lot, but they can't speak or write it very well.
Now, if you're lucky enough to grow up in a minority language region these days you'll probably be formally taught your native language from when you start school alongside your country's official language(s). There is a lot of politics tied up in language issues which I suspect puts blinkers on people instead of building bridges, particularly in multilingual regions where anything from streets to whole towns have two names in different languages and it's a big deal what you call things and which language you choose to be heard speaking in public and to whom. Not everywhere follows EU recommendations to officially teach their minority languages. The French are particularly unenthusiastic, so I have friends who speak Elsässerditsch as their mother-tongue but couldn't read or write in it to save themselves AND claim not to understand any German! Likewise I have a Breton acquaintance who only learnt to write his native tongue after taking a course in Ireland.
I suspect linguistics isn't taught very well on the whole though because I'm always amazed by how much trouble two neighbouring languages have understanding each other. Swedes struggle hard with Danish, Norwegians understand Swedes better than the other way round. Irish speakers really struggle with the pre-aspirations and voiceless plosives in Scottish Gaelic, but they'd mostly get the gist if they saw it written down (if they'd studied pre-standardised spelling). One year I had a flatmate from Valencia whose parents used to ring up on the landline. They only spoke Valencian, so my looking up phrases in Castellano did not work at all, but bizarrely, my school girl French with an attempt to match their accent actually worked to get short messages across. My flatmate was impressed at any rate. She thought I'd been swotting up on Valencian. I couldn't help but wondering if not understanding Castellano was so part of their identity that they had put up some sort of subconscious blocker to not understanding it as I really don't think what came out of my mouth was any closer! I also shared a house with an Estonian girl for a while. She'd made an effort to understand Finnish and didn't think it was hard but it didn't seem to go in the other direction.
I guess being multilingual isn't that unusual but it's most often an accident of birth or upbringing which languages ones speaks. The upside of being multilingual is that it's easier to learn more languages later because you already know that language is more complicated than substituting one word for another. Anyway, we muddle by alright. I get the impression that USians "translate" the EU into the "US of the other side of the pond." It's really nothing like that. Besides not covering all of Europe it's much, much looser. Think of it like a stronger version of NAFTA and you'd be closer to it.