I've got a sentence from a coursebook which goes: "I usually sleep early on Saturdays". What can it possibly mean?
I have to voice complete disagreement with everyone who has commented that the sentence is nonsensical despite its structural integrity. As a native American English speaker from the Northeast going to college in the South, it is perfectly common for both myself and my peers to say "sleep" in place of "go to sleep" in conversation.
As evidence, a Google Books search for "sleep early" that excludes "went to", "go to", "goes to", "going to" prefacing the phrase returns 1300+ results at the time of this post. Searching Google gives 134,000 results (of which, admittedly, many are not good examples, among which is an article titled "How to Sleep Early for a Night Owl" and an article where a commenter notes he/she does not seem to be "wired to sleep early".
I do agree that it would be technically much more appropriate to say
I usually go to sleep early on Saturdays.
as opposed to
I sleep early on Saturdays.
because it is unclear whether "early" is referring to sleep as in going to sleep or sleep as in being in the state of being asleep, particularly when the latter is read literally. Idiomatically, however, the latter is fairly commonly used and very commonly understood. The word "crash" is also used in place of "sleep" quite commonly, like so:
I crash early on Saturdays.
which means exactly the same thing. In context, "crash" may be more appropriate for emphasizing just how tired an individual is, but it carries no such connotations by itself.
The reason I mention this is because it is possible - although I have no way of really knowing this - that due to "crash", which is a much more instantaneous and less ambiguous word, being used as a synonym for "sleep", the verb "sleep" has, at least in the vernacular, come to refer to the action of going to sleep.
Per Preston's comment, OP's example is "unnatural" - and since it's not an idiomatic usage, we can't say it has a single fixed unambiguous meaning. The two possibilities are...
1: "I usually go to sleep early on Saturdays"
Meaning: "on Saturday night I go to bed early"
Native speakers would almost never omit go to in the above construction, but (rarely) it might occur in a "non-habitual" context...
“There will be music until God knows how late,” Selma was saying. “So we must sleep early tonight".
2: [whimsical reversal of] "I usually sleep late on Saturdays"
Meaning: "on Saturday morning I get up early"
Because of potential confusion with #1 above, this form would be unlikely unless there was a clear contextual contrast with the idiomatically standard late version.
If OP's example was known for certain to have been produced by a native speaker, #1 would be the most likely interpretation if it was from more than a century ago. If more recent, it's probably #2.
But given both of those are so unlikely, and taking account of other howlers I've seen cited from online coursework here and on ELU, I would say the only real meaning to be extracted from OP's citation is that his source material came from a non-native speaker. My recommendation would be to discard it.