Certainly. There is an idea that has been circulating around for a while that the Greeks and Romans didn't really value belief or faith, only getting the proper forms of ritual right, and it wasn't until Christianity that this changed. To me and most modern scholars, this is a pretty spectacular misunderstanding of both classical religion and Christianity (consider, for example, the amount of angst caused by the inability of early Carolingian clergy to properly perform Latin rites). The forms of ritual were certainly important, but they have an inflated importance for us because most of our specific information comes from a set of hyper literary intellectuals who were, by and large, not terribly devout and were mostly interested in history and literary representation. This was then picked up on by modern scholars who wrote within a colonialist, cultural evolutionist and very Christian environment, where the dictates of whig teleology demanded that the replacement of classical religion with Christianity be due to some sort of innate gap in the former filled by the latter. Still, in classical literature we have people like Euthyphro, who was famous for his piety and advocated a strong form of subordination of personal ties to universal notions of divine justice. Much later, there are figures like Plutarch, a famous second century intellectual and priest of Delphi who advocated a conservative form of devotion. Arguing from opposition, the poet Lucretius spends a great deal of space arguing that we need not lie awake at night worrying over divine justice--hardly a necessary task if there weren't people who did that. And, in a broader sense, pilgrimage sites show pilgrims coming great distances as evidence of a deeply felt popular devotion. But this is just for "mainstream" religion, when you include the so-called mystery cults engaging in intensely devotional worship such as flagellation, which I think is rather strong evidence of personal religious devotion.

As for acceptable levels of devoutness, this is hard to say, and it certainly changed over time. In the Hellenistic and particularly Roman period it was possible to show a pretty flagrantly skeptical take on religion, which is seen particularly among Epicureans like Lucretius and other intellectuals like Lucian. In classical Athens, on the other hand, Socrates was executed for religious infractions, and as much as this was a sham trial, the fact that this was an actionable charge does speak volumes (as does Anaxagoras' expulsion on the charge of impiety).

Although it doesn't directly answer your question, Greg Woolf's article "Polis Religion and its Alternatives" is a characteristically insightful and well written take on the false dichotomy between civic worship and personal devotion.