Is It Possible to Fix One Side of a Broken Nose?

If it's not broken, don't fix it.  If the nasal bone on one side only is displaced, then it is only that bone that should be reduced, at least in the acute situation.  There is no "maximum" number of times you can have rhinoplasty, and last I checked, the Guinness Book of World Records doesn't have a category for this.

It is certainly possible that only one nasal bone was fractured. A closed fractuer reduction can be done to reposition the out of place bone. The bones reset over the course of about 6 weeks so the reduction is most easily done in the first 10 days or so after the injury.

There isn't a maximum number of rhinoplasties that a person can have. With each surgery it can become more complicated and difficult, but there isn't a finite number of procedures that a person can have.