Not always possible without a lot of changes in the language. Language planning's often needed, and when you have extensive bilingualism it's hard to keep them from mixing.
The two Sorbians have had a rough go of it. A lot of German influence that the purists have trouble keeping out. It's a hothouse language. A luxury language.
Even a lot of the documentation from nearly extinct languages is suspect. You get a small group of old women as your informants and that's going to happen. Imagine if you got a small group of old American women and had to document all of American English based upon that. Make it more realistic: It's a group of old American women who have been out of touch with English-language sources for the last 25 years. Now imagine that you want to *reconstruct* a viable resurrected American English based on just what linguistics could gather from those informants in a few years of summer research. The register and stylistic skew badly. The lexicon's warped and diminished. Even pronunciation's likely to be off.
Manx is gone. It lacks the registers, native speakers. They won't coin words based on living patterns but how non-native speakers interpret the fossilized remains of the language. The lexical base is mostly reconstructed. Same for Cornish, but worse.
Hebrew's often cited as a counterexample, but if you look at a lot of the phonology and how the original grammatical categories were interpreted, it's got a lot of Slavic. Modern Hebrew isn't a direct descendent of Tiberian Hebrew. In many ways, it's relexified Slavic. Granted, that's really Wexler's POV and a bit overstated, but it has some merit. Once it was resurrected, it took on a life of its own--but "resurrection" isn't exactly the right term.
Linguistic diversity for the sake of linguistic diversity is something that most speakers don't have much use for. In any event, the linguistic "monoculture" that we keep fearing keeps not happening--and even as it happens, it breaks up. Monocultures are temporary, transient.