I got my cast and tape wet??

I would certainly contact your plastic surgeon but not to panic. I routinely remove the splint the day after the surgery and do not replace it so unless the surgeon performed a unique type of rhinoplasty, some movement of the splint at 5 days should not be a problem. 

Call your rhinoplasty surgeon for advice. I remove the splint 5 days after surgery so this shouldn't be a reason to panic. However, your surgeon might have a different approach. Let them know. Stephen Weber MD, FACSDenver Facial Plastic Surgeon

I would suggest that you contact the office which you had
the procedure done. The rhinoplasty cast typically comes off a week after surgery,
unless otherwise directed by your surgeon to keep it on longer. Since you are 5
days out it shouldn’t be too much of an issue, however I would still
contact your doctor who performed the surgery to see if
you should take it off or have the doctor reapply it for the remainder of the
days for which you should wear it.