Yes! We apply the same logic that we do to the plan fee to the time passes.
This means if 1/3 of the month has already passed, your members will pay 2/3 of the monthly fee on their prorated invoice and be assigned 2/3 of the time passes by Cobot.
If you would like your member to have access to all of their time passes regardless of when they start, you can manually add passes to their account by going to their _Member Profile » Time Passes » Assign Time Passes. Be sure to mark them as 'Don't Charge' so they are not added to their next invoice.