I see/hear many instances where the preposition "at" is omitted when a question starts with "What time ... ?" For example, I hear people say "What time are you guys meeting?" as opposed to "What time are you guys meeting at?" or "What time did you wake up?" as opposed to "What time did you wake up at?" Is it OK to assume that this "at" is optional, or am I just hearing it wrong?
The dropping of the "at" has largely occurred because it simply sounds awkward and redundant to native speakers when dangling on the end of a time-oriented question, and sounds stuffy and pedantic when placed on the front of the question, so it is avoided if the meaning of the question is clear, which it nearly always is. It's not incorrect to include it, but it is fairly uncommon (I can't find any occurrence in Google Ngrams for many common formulations)
Note that "what time" as an object of a prepositional phrase is nearly always the object of "at", as it rarely makes sense in any context to have any other preposition in that place.
In the extremely rare condition where the noun "what time" could potentially be the object of the verb being used in the question, the meaning is usually entirely clear from context. For example,
"What time are you reading?" Jan asked the poet as he left for his debut.
"What time are you reading?" Jan asked the scientist as she squinted at the timer.
If it was absolutely necessary to distinguish between these two meaning, a questioner would likely add the trailing (or leading) "at" to clarify:
"What time are you reading at?" Jan asked the scientist as she stared at the timer's display on the way to her debut lecture.