Although the period generally accepted as the end of the Roman Empire was one beset by instability, and "well off' can have different connotations, the answer to your question would nonetheless be affirmative.

Two of the most prolific writers of late antiquity, Flavius Magnus Aurelius Cassiodorus and Magnus Felix Ennodius, are positively ecstatic when Theodoric is proclaimed Rex Italiae, or King of Italy. Cassiodorus would go on to have a distinguished career in Theodoric's administration, while Ennodius to become one of the most, if not the most, well-respected cleric in post-Roman Italy.

Although two very different men from different backgrounds (Cassiodorus is a southern Italian while Ennodius is a northern Italian) they actually agree on a lot of things. And what can be gathered from their writing is that for both of them, Gibbon's date ending the empire in 476 means absolutely nothing. In fact, by the 510s Ennodius and other high-profile Italians begin writing about a sort of "Imperial Restoration." Although such optimism was, in fact, was far from the truth, what is certain is that King Theodoric was embraced as a defender of all things Roman.

But the reason wasn't exclusively because Flavius Odoacer, Theodoric's predecessor, was some sort of uncultured barbarian. To this day, we're not even sure of his ethnicity (at any rate, "Flavius" sounds very Roman). In fact, rulers of the Western Empire with mixed heritage hailing from border regions were increasingly common (the ruler Odoacer had deposed, Romolus Augustulus, fits this description). Rather, for these writers, Odoacer was only the latest bad ruler in a succession of bad rulers.

But when the Consul Theodoric was sent by the Eastern Emperor to stabilize Italy, his success or welcome was far from a foregone conclusion. In fact, many previous writers had defended leaders of "barbaric" origin as being more "Roman" than the "Greeks" sent to help, through the years, by Constantinople. I actually wrote an answer about ethnicity and its role at the time of the foundation of the Kingdom of Italy yesterday.

As much as I dislike "Great Man" histories, and think that too much scholarship on Italian Late Antiquity focuses solely on his perspective, there is little to disprove that Theodoric was an astounding man. Even before his conquest and usurpation was complete, the Italian aristocracy absolutely fawned over him: he was cultured, refined, intelligent, but also possesses a military virility in keeping with the "Old Roman" values. He also extensively taps the Latin upper classes for important positions of power.

Ennodius was one such person. He was born in southern Gaul as a descendant of one of the oldest Gaulish Senatorial families. When the administration in Rome had finally given up on his homeland when he was still very young. Although most Romano-Gauls more or less seamlessly albeit begrudgingly acknowledged the fact that they had been living in a completely autonomous region run by a Frankish warrior elite that was nonetheless very Romanized (Sidonius Apollinaris being the primary example; but I digress) the highest tiers of society like Ennodius' family elected to relocate to Northern Italy (Ennodius, later in life, loved reminding people he was related to everyone who was anyone).

Ennodius retained social and familial ties in both sides of the Alps, but his impressions of the world are profoundly influenced by his formative years in Northern Italy much more than they are by his Gaulish roots. Not only was he educated in the latin classics, but growing up witnessing long imperial sojourns in Mediolanum (modern Milan) and knowing that the Imperial court, more often than not, had a permanent residence in Ravenna, down the Po river, led Ennodius to develop very specific and traditionalist views of what qualities made for a true Roman. In addition, although he probably never set foot in Rome in his youth, Ennodius nonetheless also ascribed to the city profoundly symbolic and spiritual qualities.

But in spite of his persona as the truest of true Romans, Ennodius did not live in denial of the fact that the Empire he idolized was in a state of deep crisis, and had been long before he was born. The "Empire of Italy," as he was wont to refer to it, needed a ruler worthy of it; Theodoric was that man. Over the course of Theidoric's reign, Ennodius starts his career as a deacon working for the Bishop of Milan, and eventually climbs to the post of Bishop of Pavia; in the meantime becoming one of the greatest ideological narrators of Theodoric's rule and participating in important diplomatic and ecclesiastic missions to Constantinople.

But Ennodius was far from an exceptional case. Flavius Magnus Aurelius Cassiodorus Senator is perhaps an even more illustrious example. Cassiodorus was born to a family of extremely prominent Greek-speaking Southern Italian Patricians. His father had taken up important positions in Odoacer's administration, but he evidently wasn't so close to the previous King that Theodoric felt the need to purge him (although as stated before Theodoric was generally very magnanimous with members of old Roman families). Cassiodorus' father is nonetheless sent to govern Sicily, where he'd be rather harmless, and was later moved to his native Calabria for his last years of service before retiring. Cassiodorus picks up where his father left off, and governed Calabria for a few years until Theodoric calls him to Ravenna. Theodoric nominates him Consul, but then he generally drops off the map, presumably climbing the ranks of Theodoric's administration until he became Magister Officiorum: literally the "Master of the Offices" (effectively Theodoric's Prime Minister).

After Theodoric's death and in the subsequent power struggle, Cassiodorus does his best to keep the administration running as smoothly as possible. But he seems increasingly ineffective, and is almost trapped in Ravenna. After Justinian reconquered the peninsula, Cassidorus drops off the map. There is evidence of a bit of itinerancy, sometimes in Calabria, other times as far as Constantinople. He eventually founds a monastery near his hometown of Squillace, where he would end his days.

But these are all people who, while starting from privileged positions, certainly saw their lot improved and their skills rewarded by King Theodoric and his stable rule. Were there people who served Romolus Augustulus, Odoacer, and Theodoric? Well I don't know of any person who did, but I do know of a dynasty who did: that of Anicius Manlius Severinus Boethius. Boethius' father Manlio had been prefect and consul under Odoacer, his grandfather prefect and consul under Valentinian III, and it's also probable he was related to other Severinii who had been appointed to important positions in the second half of the fifth century. His sons, in turn, would both be consuls in the 520s.

After an extensive education in Athens and Alexandria, Boethius dabbles in translations and philosophy and Theodoric eventually taps him for his administration. However, the reason Boethius' dynasty is a more interesting example, as a whole, than any given individual's is that Boethius' story doesn't end well: he's accused of treason (possibly on trumped charges by rivals in the Senate) and eventually executed. His father also died under unclear circumstances during Theodoric's campaign against Odoacer.

Still, I like Boethius' example because it's proof, on the human level, of how an old Senatorial family could still manage to send their son to study in classic centers of learning in spite of the upheavals of the fifth century, and that son could have a fairly traditional career dallying with philosophical writing until tapped for administrative positions. And personally, although modern academia has long since discard the idea that there is a definitive break between the ancient world and late antiquity, I'd say I'm on the further end of the spectrum arguing for a more seamless post-Imperial societal continuity (not entirely unaligned with historians like Patrick Amory, although he's on the furthest end of that argument).