I don't know about "ability to purchase sweets" as a cause, but there are certainly records, albeit in satire, of people (noble or otherwise) having mouthfuls of rotten teeth. Martial V.43, for example, gives a satirical account of a pair of citizens, Thais and Laecania--one of whom has a mouthful of rotten, black teeth, and the other of whom has had to purchase an entirely new set of dentures (hence, her teeth are white). It's just a pithy elegiac couplet, so there's really no cause mentioned as to what happened to Laecania's poor teeth.

Martial V.43:

Thais habet nigros, niveos Laecania dentes.

Quae ratio est? Emptos haec habet, illa suos.

Quick, loose translation:

Thais has black teeth, Laecania white; why is this? The former's teeth are her own, while the latter's teeth were bought.

A cause I have seen cited for the rotting of teeth in the ancient world, though I don't know how much historical weight it pulls, is uric acid--people would supposedly "brush" (read: clean) their teeth with urine, and even the most durable bodily substance is compromised by that kind of acid, eventually.

A quick search of references on Martial's satire didn't yield much in the way of clarification for me, but I can say for certain that, yes, ancient people were subject to tooth decay based on diet and hygiene, and, yes, there were people whose teeth rotted and who could then afford to buy pearly-white dentures. And they did, because, well, the Romans were rather obsessed with things like white teeth, healthy hair, signs of physical wellbeing in general.

EDIT: Fixed typos.