The clusters in "upthrust" and "backstop" actually have three consonantal sounds

Yes. I wasn't intending them as examples of more than three, but of counterexamples to the rules that DanielLC proposed.

Those clusters aren't in a single syllable.

The original comment didn't talk about syllables.

I haven't seen more than three consonants in a single syllable.

"Firsts." On the other hand, a phoneticist might analyse the "ts" part as a single sound; except that on the phonetic level it appears to be two phonemes. So is (the sound represented in English spelling by) "ts" one consonant or two? Is the answer different for "tsetse" and for "firsts"? For "Katz" and for "cats"?

Linguistic categories are complicated.