There are plenty of claims, but there is no way for modern historians to confirm any of them. The Anicii only appear in 298, when Anicius Faustus became the consul and I don't think there's any indication that they publicised their Republican heritage - the late antique Anicii originally came from North Africa, whereas the Republican Anicii came from central Italy. Of course, the imperial Anicii could still be descended from the earlier family, but we cannot evaluate this hypothesis with the available evidence. The Anicii's claim to antiquity may have come from their marriage into the Acilii Glabriones, who did have what appears to be a long and distinguished ancestry. But even then this is only based on the fact that a certain Anicius Acilius Glabrio Faustus (consul in 438) was the son of Acilius Glabrio Sibidius, which has been interpreted by Alan Cameron as evidence that Sibidius had married a female member of the Anicii and added 'Anicius' to his son's name to 'advertise their union'. The Acilii Glabriones claimed descent from Manius Acilius Glabrio, the consul in 191 BC, and their claim was seemingly well-known, as the third-century historian Herodian discussed another member of the family, Marcus Acilius Glabrio (consul of 186)

This Glabrionus was the most nobly born of all the Roman aristocrats, for he traced his ancestry to Aeneas, son of Venus and Anchises, and he had served two terms as consul.

Cameron however points out the obvious problem, since we know of no-one from this family between yet another Marcus Acilius Glabrio (consul of 256) and Sibidius in the late fourth/early fifth century (though there is an unknown Acilius Glabrio from an inscription in the early fourth century). For a family to pass their name through their male line for five centuries is extraordinarily unlikely (see for instance the often-noted longevity of the House of Capet in France - which is only frequently pointed out because it was so exceptional), so it would be safe to presume that descent from a female line and/or forged ancestries were involved somewhere along the way.

There were of course many other claims. Lucius Aradius Valerius Proculus Populonius, prefect of Rome in 337/8, was for example praised by a friend to be descended from the Poplicolae family active during the Republic, whilst the fourth-century inscription of Creperius Amantius and Caeionia Marina declared their descent from Munatius Plancus Paulinus, consul in 13 AD. The Decii and Corvinii, both families active at the same time as the Anicii, similarly had their Republican ancestry praised by Cassiodorus and Ennodius, two authors writing in Ostrogothic Italy in the sixth century (interestingly, the Anicii were not praised in relation to their Republican ancestors here). Personally, the most fascinating theory is the idea that Emperor Anastasius (491-518) was a descendant of Pompey the Great, as the Republican general was mentioned a few times in panegyrics dedicated to the emperor and the name 'Pompeius' is attested within the emperor's extended family - which is an indication that the family held the name to be somewhat important to pass on through the generations.

Understandably, despite all these claims, we have to be cautious, since we are reading what these families and their allies want us to read, not factual reports of their descent. The Romans loved everything from the past and evidently fictional ancestries were omnipresent. St Jerome for instance celebrated his patron Paula's descent from Agamemnon from the Trojan War, whilst Ruus Volusianus was allegedly descended from the Volusus featured in the Aeneid. Most famously, Constantine the Great (306-337) manufactured an entirely fictitious familial connection with the relatively recent emperor Claudius Gothicus (268-270). There is no reason for us to think that other Roman aristocrats were not capable of writing similar fabrications. Many people no doubt believed that they had many illustrious ancestors, and in reality they probably did, as they no doubt were all descended in some way from Roman families dating back to the Republic, but we have to be very cautious about their claims that they were descended from specific individuals, since the evidence available is not enough for us to test any of them.