Oh yes, you can overbrine a turkey.
One of the main issues with brining poultry for long periods is not keeping it cold enough and bad stuff grows. If you keep it just above freezing, which is easy to do since the water is salty, no problem.
Now, the amount of salt in a brine can make the meat too salty. The standard brine formulas call for about 1 1/4 cups of Kosher salt per gallon of water used. When I first started brining birds a LONG time ago, I used that amount of salt on a fresh bird and brined it for less than 24 hours. Wife refused to eat the turkey. Said it was too salty. I thought it was okay, but it tasted salty to me too.
I've cut the salt to 1/2 cups per gallon, half that if brining a pre-brined turkey (aka Butterball) in OJ or Apple Juice. I can easily brine a bird for 72 hours or more without it being too salty. Just did one yesterday, brined 72 hours in OJ, water, and spices. The juices just flowed out of the white meat it was so good. I put 1/2 tsp to a full tsp of cure #1 in my poultry brines which gives just a hint of a ham flavor that goes well with the smoke. I also inject the brine into the breast.