When you get a shot and end up with a big bruise and they say it's because you bruise easy, is this the case?

Some individuals do in fact bruise easily, they may have an ascorbic acid(Vitamin C) deficiency or some coagulopathy. Perhaps they are taking a blood thinner medication like Coumadin, ASA, or Lovenox.

Anytime the tissues are traumatized bruising is possible. Receiving a shot IM or SQ is trauma, albeit very little trauma, nevertheless still trauma. Unfortunately though, a common cause of bruising after receiving an injection is poor technique by the person that administered  the shot. Blood slowly leaks into the subcutaneous, dermal and epidermal tissues and causes a bruise.