Yes, they did. The conspirators who assassinated Emperor Caius (Caligula - Augustus' adoptive grandson) on 24 January, AD 41 (27 years after Augustus' death), endeavored to slaughter the whole imperial family. The plot was led by none other than Cassius Chaerea, the commander of Caius' Praetorian guard (the personal guard of the emperor), with the help of several senators. If the entire imperial family were dead, he surmised, then the army, the people, and the pro-imperial faction in the Senate would be forced to accept a return to senatorial rule.

Unfortunately for the republican conspirators, certain members of the Praetorian guard whisked Claudius off to the safety of the Castra Praetoria, the military headquarters of the Praetorian guard in Rome. The Praetorians hailed him as their new emperor. Claudius refused to leave the Castra Praetoria until the Senate was willing to acclaim him as their new leader, which they did.

Some Romans believed Claudius was involved in the massacre, and that the plotters had lied to him about their motives; that is, that they had convinced Claudius that they were going to install him as Emperor when, in reality, they were going to kill him as well the entire time. Others believed that Claudius really was the victim of a series of coincidences that allowed him to survive. Josephus even makes the claim that Claudius acted almost entirely on the advice of his childhood friend, Herod Agrippa, who would later go on to be ruler of Judea, and would otherwise have bungled his way into death.

Sources: Suetonius, Cassius Dio, Josephus

Something closely related to your question occurred at the end of the reign of Claudius' adopted son Nero. Since the two events happened so closely together, as it were, I want to mention this event as well. In AD 68, a revolt against Nero's tax policies occurred in much of the West, and several important people declared themselves in support of a new emperor. The specifics aren't important, so we'll skip to what matters, which is this:

Nero fled the city and attempted to reach Ostia, the port of Rome at the mouth of the Tiber, so that he could flee to the still-loyal East. The Senate was cognizant of the fact that Nero was the last living member of the imperial family, and - for that specific reason - hoped to guarantee his personal safety, his life, so that an heir could be produced. Not because they liked Nero as a person. Not because they supported Nero over the rebels in the West. They wanted to save Nero to protect the imperial bloodline. Might they have pushed for a quasi-constitutional monarchy?

We'll never know. When the Senate's messengers caught up with Nero, he assumed that they had come to kill him. Rather than be executed, he commanded his private secretary to murder him when the sound of the messengers' horses grew too loud. We can't say what might have happened had he listened to their offer.

Source: Tacitus

Off-topic: For those of you who are interested in seeing a film adaptation of the struggle between republicanism and monarchism in the early days of the Roman Empire, and who want to see the kind of messy dynastic politics that Augustus and his relatives experienced up to the death of Claudius, I would recommend the inimitable 1976 BBC television mini-series, "I, Claudius", adapted from Robert Graves' novels I, Claudius and Claudius the God. Both the books and the show are entertaining, fanciful re-imaginings of the history of the period - not documentaries, by any means - but the broad strokes are there: the names of the important persons, the key events in their lives, and the outcomes of their actions. Again, I must stress that this is a work of entertainment, not a historical documentary. But if you have an interest in learning more about this period, and if you enjoyed HBO's "Rome", I'm sure you've always wanted to see a television series about what happened after Augustus took over.