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My question is actually a bit more broad than what's in the title, but I don't know how to put it succinctly. When I was trying to find the answer to that question, I found that all known terrestrial vertebrates are ultimately descended from a species of lobe-finned fish that had 2 sets of bilaterally symmetric fins (which excludes the dorsal and ventral fins), which were the basis for the limbs of its tetrapodal descendants.

The best way I can think of to phrase the question is: Does paleontology has any evidence of a vertebrate that had 3 (or more) sets of bilaterally symmetric limbs or lobe-based fins? I'd like the focus to stay away from ray-finned fish, even though they're vertebrates and related.

This is not the common question, which is whether there is or can be a mammal/reptile species with more than four limbs. The essence of that question's answer is that mammals and reptiles both evolved from a common ancestor species that were tetrapods, and that it's extremely difficult to change the genetic floor plan in the necessary way without causing fatal problems (along with other evolutionary issues). By and large, this common question is so common that it chokes out the type of question that I have, making my amateur attempts at further research difficult to impossible.

TangentialThinking
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I've not read it but this SE question Why don't mammals have more than 4 limbs? though not a duplicate (it asks about mammals not all vertebrates) may have some elements in its answers pertinent to your question.

And the question Why are there no vertebrates with more than four limbs? has been asked in Quora where it has 7 answers you may find useful.

Also this discussion thread Why can insects have 6+ limbs, but mammals only have 4? in reddit.

Does paleontology has any evidence of a vertebrate that had 3 (or more) sets of bilaterally symmetric limbs or lobe-based fins?

You sent me far down the rabbit hole with this one, for reasons that escape me my instant response was 'yes the Coelacanth' but as you correctly point out that has only 4 paired fins.

As seen from The origin of tetrapods our limbs came from fins, Coelacanth is a good example of the likely early beginnings of Tetrapod limbs, they occasionally use them to 'walk' on the seabed.

The image provided in this National Geographic article Coelacanth Rediscovered gives a much clearer picture of it's 'limb' structures.

Coelacanth

Latimeria chalumnae, the coelacanth (SEE-lee-uh-kanth), thought extinct until a biologist identified one off the coast of South Africa in 1939.

I've trawled Google for early chordates looking for examples of more than two pairs of fins & haven't been able to find any, it seems the earliest paired fins appeared in (or shortly before) Ostracoderms & from there continued in an unbroken line to the tetrapods & our limbs.

Paired fins don't appear to have developed in chordates before then or to have ever developed into more than two pairs.

So to the best of my knowledge it appears that the answer to your question is no.

Other links. Yunnanozoon Myllokunmingia Haikouichthys Evolution of fish

Far later edit: I "might" have found one?

Came across this article on Polymelia which says "Mesacanthus developed a third pair of paired fins" // & this article on Mesacanthus I then found says besides the normal paired pectoral & paired pelvic fins it had "a pair of prepelvic spines".

Pelinore
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    "there have never been any land vertebrates with anything other than four limbs." - snakes. If there are any vestiges of limbs on their skeletons, I haven't been able to spot them. And there certainly isn't anything on the outside. – John Dvorak Sep 07 '19 at 18:40
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    @JohnDvorak : Snakes were just 4 limbed lizards b4 losing the last vestiges of their limbs, they are for the purpose of this question four limbed animals I feel, you can even prod the embryos hormonally to grow limbs if you want, it's all still in their DNA they just don't express the genes, but point noted :) – Pelinore Sep 07 '19 at 18:50
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    While the coelacanth has 6 fins (along with many related extinct species), only 2 sets have bilateral symmetry, the remaining 2 fins are unpaired dorsal and ventral fins. The information in the links has been interesting, though I'm still combing through it. – TangentialThinking Sep 09 '19 at 03:06
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    I would shy away from pointing people toward Quora or reddit since the answers on both sites are often not factual, opinion, and lack adequate peer review for accuracy. Referencing such sources of information is, in my opinion, reckless and very potentially misleading. – theforestecologist Aug 20 '21 at 13:03
  • @theforestecologist I'd more or less agree with you, particularly about reddit, but they do occasionally provide links (or even just technical words I hadn't thought of useful for a search) & can make a useful jumping off point as a result. – Pelinore Aug 21 '21 at 15:17
  • @Pelinore that may be true, but I think relegating such sources to minor footnotes would be more appropriate than citing them in such a front-and-center fashion as in your post. (e.g., something like "you can find similar posts on less-reputable sites like Quora or reddit to use as further jumping off points"). Again, I think information on both of these sites are notoriously dubious, so I would personally suggest/request you help to slow the spread of misinformation by not promoting those sites as prominent parts of your answers here. We're trying to maintain higher standards here at SE :). – theforestecologist Aug 21 '21 at 20:57