With tieflings, that's definitely the case. I'm not so sure about dragonborn.
I still have my Wizards Presents: Races and Classes, the thin book on design thoughts they put out in the lead up to 4e. Richard Baker talks a little about how they selected the tieflings and dragonborn. With tieflings, as you say, he calls out their enduring popularity in 2e and 3e as the reason for their promotion to the PHB. The dragonborn, though... let me quote.
"Once we had this list, we asked ourselves the question, "What's missing?" Could we think of any iconic fantasy races or cultures that hadn't been presented in the 3rd Edition and needed to come forward? We briefly kicked around the idea of the "talking animal" race: after all, the Narnia movie was pretty good, and talking animals show up all over the place in fantasy literature. But we suspected that the mechanical design of such a race (or collection of races, really) would be very difficult, and we were afraid that most folks in our audience would see talking animals as some kind of bad joke. While that bit of brainstorming didn't work, some additional brainstorming did. We realized that we had several varieties of "dragon man" rattling around in the system, and that we might combine them into a single character race with its own unique culture, society, and mechanical niche."
I honestly think they missed the mark by trying to force the issue, here. Half-dragons were the most popular of the 3e "dragon men" types, and they had a very different aesthetic and story from dragonborn. Also they may have tried to hit those last points in 4e, but in 5e they certainly don't have a unique culture, society, or mechanical niche.