There are various issues that can happen. Most of them depend on the type of materials the stuffies are made. The older the stuffed animal, the bigger the chance of more brittle stuff was used, like straw stuffing. Also back in the 60s and 70s the materials were not that long lasting.
For example the fur can start coming loose and falling off. Or the stuffing can condense and lose form. Details like eyes, eyebrows and whiskers can also call off. Tears in the main fabric can appear or the stitching can become loose.
The good thing is that you can tackle most of these problems relatively easy. You just need to be patient and careful.
The teddy bear you see in the main picture is about 30 or so years old. It has the old type of stuffing that gets condensed with time, so the bear looks soggy now. It is in need of a change of stuffing. The easiest way to to that is to rip the seam open somewhere on the side of the toy and remove the old stuffing. Then use polyfil, foam or cotton for the new stuffing and sew it close.
But what if you don’t want it to come to a need for repairs? You can dress up your stuffie in a shirt which will help protect the fur and body. When handling the toy, don’t squeeze it and throw it around. Instead be very careful and gentle.
Also make sure the conditions in the room you keep the toys are good. There shouldn’t be too much moisture in the room. Keep the stuffies away from direct sunlight as it can cause the colors to fade over time. If you keep the toys in a cabinet, make sure there are small openings here and there to allow air to circulate.
These are some tips that will help you preserve a really old stuffed animal and keep it in good condition for much longer.