Context: I have a small linux computer which isn't in use, with the potential to add others. However, I'm worried about excessive data transfer slowing down my internet connection when I am using it otherwise.

Question: Is it possible to schedule the times during which the computer will access the internet? I.e. to do most downloads and uploads from 2-6AM?

Yes. Open BOINC Manager, and switch to 'Advanced View' (if you aren't using it already).

From the menu, choose 'Options', 'Computing preferences...'. Switch to the fourth tab, 'Daily schedules'.

In the lower part of the screen, fill in your preferred schedule, and save it.

Also look at the 'Activity' menu. Make sure that 'Network activity according to preferences' is selected.

That should do it.