Yes, it is totally program dependent. It really all depends on their surgical volume and number of residents.
At some clerkships, you'll barely scrub in at all (since it's easier to see if you're not scrubbed in if 2-3+ residents are already scrubbed in).
At others, it will just be you and the resident, and you will get a fair amount of exp depending on the teaching/comfort of the attending and the resident.
It is a bit of a paradox since the "first year cases" like HWR, amps, I&D, digits, etc will generally let you do more due to only one resident there (if it's a quality program), but the more complex cases are more interesting to watch (yet you will be blocked by multiple residents/students to some degree).
Also, don't ever take it as a knock on you if the attending and/or resident fly through the case and don't let you be involved much. Sometimes they need to get back to office, have multiple cases to do afterward, the case started late, etc. You don't realize it when you're a student, but even if you have practiced your sutures, etc... you are still probably in the bottom 10 percentile among attendings and senior residents.
...Your best bet for getting the most of cases is to read about the the night before. It helps to ask the resident what fixation will be used, etc. You might get a basic question or two on the procedure or the setup (TAGS-X stuff), and the right/wrong answer could increase/decrease your involvement. I usually did that for students or a resident I wasn't familiar with