In theory yes, although shoddy record keeping makes many claims that are difficult to substantiate.

Although the Visigothic Kingdom in Spain, Vandal Kingdom in North Africa, and Ostrogothic Kingdom in Italy were fundamentally run by a very romanized ruling class of Germanic origin with a predominantly unchanged Latin bureaucracy, there are some specific instances of important families in Western Europe claiming descent from Roman patricians.

The Visconti of Milan would have us believe they were Romano-Celtic patricians living on the shores of Lake Maggiore since the aftermath of the Trojan War. However don't think the sources are airtight, and they're questionable and propagandistic, many conveniently having appeared when a match was being made between Gian Galeazzo Visconti and Isabel of Valois, daughter of the King of France. The most complete history of the Visconti was written in the 14th century by the Milanese chronicler Galvano Fiamma, who spends a great deal of time on the Visconti's great lineage in his Politia Novella, or modern history of Milan, today stored in the Ambrosian Library. The tone he takes when speaking of the Visconti is unsurprising given he was Duke Giovanni Visconti's personal chaplain.

Examples go on: the House of Este, which ruled Ferrara and Modena (and who are related to almost every other Royal house in Europe) are themselves are a Junior Branch of the House of Welf, following the 11th-century union of the Elder House of Welf (who were themselves a junior branch of Burgundian Royalty) with the House of Obertenghi, who were Lombard Nobility claiming descent from the Roman Gens Attia.

Roman descent was also popular among non-dynastic families, especially in the Republic of Venice. Dandolo family, for example, traces their lineage back to the Roman gens Ursia, specifically from Orso Ipato and his son Teodado Ipato, who were Byzantine Military officers when the Northeastern Italian lagoons were a Byzantine protectorate. Although the authorized Dandolo family tree finds a way to work around it, I have to point out that there is a two-hundred year gap in the paper trail between Teodado Ipato and the next Dandolo, one Domenico; a Venetian merchant active in Costantinople unremarkable apart from having somehow smuggled the remains of St. Tarasius from Costantinople to the Basilica of St. Zaccaria in Venice.

The Badoer family of Venice also traces their origins to the Gens Ursia, and in the early tenth century there is indeed record of one Ursus Badovarius applying for a subsidy when raving Maygar hordes devastated his land somewhere near Altino at the edge of the northern lagoon. That area between the mouth of the river Sile in the northern lagoon and the fortified town of Eraclea was one of the primary settlement areas for Byzantine soldiers, so I suppose the claim is realistic, but really we're taking the Badoer on their word here. The examples among Venetian families go on; were the Contarini descended from the Roman Aureli Cotta as they claim? We don't know, the first we hear of them is a jurist in the early ninth century underwriting a will. And the Giustiniani trace their heritage to Emperor Justinian! We have spotty documentation of a Byzantine family called Giustiniani in Istria that arrives in Venice sometime before the eleventh century, but that's it.