What could cause a language developing a case system to create the singular and plural makers of a case from seemingly unrelated sources, such as in Indo-European?

I've taken an interest in Proto-Indo-European and its morphology lately. Some PIE plural case markers are reconstructable as the singular case ending + plural marker: The nominative s/es (athematic) and os/oes (thematic), accusative (o)m/(o)ms, and maybe the locative (o)i/(oi)su. But others seem completely different, such as the instrumental with -h₁ in singular and a mish-mash of -ōys, -bʰis, and -mis in plural, or dative with singular -ey or ōy versus plural -(o)mós and (o)bʰós. Those definitely don't look like a plural marker on the singular.

So what happened? Why, after a particular particle became gramatticalized as a case marker in one of singular/plural, did the speakers of (pre-)proto-indo-european not simply change the grammatical number normally instead of finding a whole 'nother particle?

Are there any other documented cases of case systems forming where the singular and plural markers differed this much?