Well, Euphorbia was brought up so I guess now I have to comment! :-)
I’ve been in favor of breaking up Euphorbia for a long time and Chamaesyce, Pedilanthus, Monadenium, Synadenium, Tithymalus, and others have been recognized. This taxonomic battle has been going on since at least the early 1800s and became quite a back and forth during the 1900s with various experts either deciding to lump all but the more divergent forms together or split out dubious groups. Few considered all genera that are now in Euphorbia to be in one genus but preferred subfamilial taxa like tribe or subtribe (see Euphorbiinae). There are several things we now know that get in the way, though.
The most basic way to explain the situation is that you have a body plan shared throughout much of the genus with many alterations from and reversions back to that form (viz. subg. Esula and sect. Nummulariopsis) in all four major clades such that there is little cohesion except at monophyletic tips where noval characteristics have developed (like C4 photosynthesis Chamaesyce/sect. Anisophyllum or spurred cyathia in Pedilanthus/sect. Crepidaria). In other words, you can recognize the noval groups at genus level, but only if you recognize two groups that have very similar body plans as different genera too. In otherwords, if the more conserved members across three of the four clades went extinct, you might be able to split them easy enough. But with them extant, you can’t really do that cleanly.
A more familiar example is the problem of recognizing birds at the same taxonomic rank as reptiles when birds “are” reptiles in that they descended from and are nested within reptiles. The phylogenetic term is paraphyletic. Personally, I’m one of a those weird taxonomists that think it might be worth accepting paraphyletic taxa, but the vast majority frown on that position.