They do go bad but most fail outright so it's probably rare for one to be "off" so much. It's more likely the limit switch is positioned closer to the heat exchanger than your probe was. Try drilling a hole in the limit mounting plate & position your temp probe so it's touching the face of the limit, then re-insert the limit and see what your reading is then.
Have you had your furnace cleaned lately? Your blower could be gummed up with clinging dust, maybe you have many registers or dampers closed or blocked with furniture. The furnace needs to breathe or the plenum temperature will rise.
That said--I have that issue (inadequate flow) and don't know what to do to correct it.