No, 1 and 2 don't imply that it was the same bone each time. In fact, they rather suggest the opposite. As others have pointed out, in mentioning fractures, one usually is specific about the bone, or at least which part of the body the bone is in:
She has a hairline fracture in her elbow.
Since your first two examples avoid such specificity, the listener would tend to assume that the unfortunate person you're discussing has had three separate mishaps recently, each resulting in a different bone being broken.
Your third and fourth sentences are unidiomatic. "A different bone" makes sense only in a context where you say something like:
Last year I broke my tibia, but last week I broke a different bone.
It can't be used to mean "three distinct bones", as you want.