"I do not know your university schedule."
Translation:Senin üniversite programını bilmiyorum.
Yes, when you leave it out it becomes
Üniversite programını bilmiyorum.
this might also mean
(Onun) üniversite programını bilmiyorum. (I don' know his university schedule)
(Ben) üniversite programını bilmiyorum. (I don't know the schedule of the university)
If it was sizin, than would it be programınızı? I guess 'üniversite programı' is a compound noun, the -nı is the accusative suffix, but I can not see the possessive suffix of senin. Can someone say another example with a compound noun ending in a vowel?
You can only have one possessive suffix on a word at a time. Ergo, if you have a possessive pronoun before something, you no longer have to form a noun compound (which is just a form of the possessive suffix). Basically. "Senin üniversite programın" + "ı" = the phrase
And you were right about "programınızı"
So the right splitting is not
program-ı-nı (noun + compund possessive suffix + accusative suffix), but
program-ın-ı (noun + possessive suffix of senin + accusative suffix).
For me it's clear now, thanks Alex!