Hiya!
Sure, why not? As one or two others have said, "back in ye olden days of yore..." (re: 1e), leveling was a LOT slower. When you hit about level 4 or 5, after playing weekly for around an equal number of months, getting to that next level was pretty much like starting a new character from scratch with 0xp. The XP amount needed would pretty much double at that point (like, 16k to hit level 5...now you need 32k total to get to level 6; depending on class, obviously). The point is...I'm used to, and prefer, that speed of leveling. IMNSHO, it should take about a year of 8hr/wk play to hit about level 9 or 10. After that, gaining 2 levels per year, roughly, is a good speed. To hit level 20 SHOULD take about 6 years or so. That's how long it took me to get my 1e MU to level 20 (and my friends to get their fighters to level 22...although one was level 16 at the end of the 'campaign' due to a rather unfortunate run in with a lucky as hell vampire...but that's a whole other story!).
As for a 5e campaign with "no leveling"...I think it would work just fine, mechanics wise. I mean, start everyone at X level with equipment and cash equivalent to a 1st level character. Now you have motivation and a sense of 'material accomplishment' as they advance. Being able to buy better weapons, armour, pick up horses, hirelings, wagons, 'luxury' camping equipment (pavilions, portable bathtub, etc)...all that would give the player a sense of "getting better".
With regard to "doing an AP like Storm Kings Thunder"...no-leveling makes even more sense. One of the, if not THE, most vile thing that an AP does is 'force' the PC's to level up. Just ONCE I'd like to see a full-fledged 'Adventure Path' start PC's at level 4 and end at level 7. IMHO, just about every single AP ever produced would have benefited from having a MUCH smaller level range. If they were done right, you could have even tied multiple AP's together to form a grand campaign story! But no.

Today's "gotta have it NOW" mentality would have a conniption fit if they played the same 7th level character for 10 sessions in a row and didn't gain a level. Too bad...
Yes. I'd play. Sign me up!
^_^
Paul L. Ming