There's two interpretations as I see it, both picked up on here: that he is referring to languages you don't necessarily 'know'; and that he is also referring to how an individual may speak a language with a very particular dialect.
In my own experience, I fell down with French, the verbs are utterly alien and I'm unable to pronounce it without feeling like a prize prat, but I can muddle on with German quite happily, it's almost instinctive (though probably grammatically incorrect). I have put that down to it being much closer to English, and also easier for someone with a northern tongue to speak. Northerners speak with an open mouth and get their tongue around words (as though munching a meat & tater pie
), much as you do with German, whereas a lot of southerners talk 'far back' and swallow half the letters rather than speak them - this is more suited to pronouncing the Romance/Latin languages.
As for dialects and accents, I use both. I find I try to speak more clearly when dealing with people who speak RP, unless they annoy me, in which case my obstinate genes kick in and I slip back into broad Lancastrian
I have perfect understanding of some of the most difficult British accents but struggle with anyone who talks far back. English has hundreds of varieties, even within England. If this is what Tolkien means, i.e. the difference between standard English and regional English, then I know exactly what he means.