If you don't get an answer here, do try over at /r/linguistics. Many of the flaired users there are also active here, but in case no one sees it, it's probably worth reposting.

When I was in college, and in graduate school, the standard answer was "Of course, not, stop being crazy." Now the answer has shifted to "yes."

I'm going to give you more information than you've asked for, because someone might find it interesting.

When we establish a relationship between languages, we do so (traditionally) by lining up words in two or more columns and comparing them, and looking for similarities. We call these Swadesh lists.

If you know a bit of English and Spanish, you can do this yourself:

One Uno

Two Dos

Three Tres

Four Quatro

Already, a linguist, or even an amateur, would be pretty sure that these languages are related. It gets harder, though, when the languages descend from a common ancestor farther back in time.

When historical linguistics and linguistic reconstruction became a thing, it was already known that Latin was the parent language of Spanish, Italian, French, Romanian, etc, and that these were closely related sister languages. That evolution happened during the course of written history, and we have a record of what happened.

But there came about talk, as early as the 1600s, culminating in an ambitious attempt in the 1900s, of unifying the Romance, Germanic, Slavic, and Indo-Iranian families (among others) into a superfamily. Since these language families split apart well before the historical era, we had to rely only on our parallel columns, and a few other tricks we discovered along the way (there are, we discovered, patterns in the way that sounds evolve, so that what is a "P" in one language is often an "F" in a sister language, or what is an "F" in one language is a "Th" in another.) Ultimately, though, by the 1900s, we've got this Proto-Indo-European thing pretty squared away (the similarity was, after all, salient enough for you to see it in the English/Spanish number columns), and people started gunning for bigger super-families.

Rigor kind of took a back seat.

People got up to this "Nostratic" hypothesis that tried to unify Japanese, English, Tamil, Hebrew, and most other things under one language. They started reconstructing a Nostratic vocabulary, even. The current global consensus is that it's bullocks, and the alleged similarities between these superfamilies are no better than you'd find by chance.

It kind of got to be taboo, from then on out, to say "HEY, I FOUND SOME SIMILARITIES BETWEEN THESE FAR-FLUNG LANGUAGES." It's analogous to someone saying they found a lost tribe of Israel in the ancient Americas. You've heard it before. You turn off the TV.

And so in 2008, when some guy named Edward Vajda proposed that he had connected the Na-Dene languages (of which Navajo is the most commonly known) to a group called the Yeniseian languages way in the Siberian interior by lake Baykal, I read about it and thought "God, not again. Idiots."

Turns out, it pans out. Experts on the languages have cautiously come over, one by one, to the superfamily, Dené–Yeniseian. Here's a Swadesh list: https://en.wiktionary.org/wiki/Appendix:Den%C3%A9-Yeniseian_Swadesh_lists

SO, it looks like we've got a trans-Bering language connection that somehow hasn't been washed out over all that time. And it looks like the greater Navajo family have found their nearest of kin, out towards Tunguska.

I love life.