Some of the boards in my hardwood floors (1949) are disintegrating.
Can anyone tell me what gives? And how I might go about stopping it?
Its either rot or termites.
The discoloring suggests a water leak that persisted long enough to rot. Do you have access under that spot? Is the subfloor similarly discolored? If so, it may need to be repaired at the same time as the board.
Board replacement is possible by any flooring contractor by sawing out the center of the board and chiseling/prying out the side pieces. The replacement piece will be dropped in by cutting off the lower half of its groove (of the tounge/groove pair) and glueing it in. Small finish nails can also be used to tack it down (and set and filled with colored wood putty).
The new piece may be a bit proud if the floor has been refinished more than once, and can be presanded to blend.
Here's a video found by Shirlock Hoimes on the TOH site
Looks like water damage. Did it occur while you owned the house? If so, and you don't remember any flooding, then the water is coming from underneath the floor. This needs to be fixed or you will continue to get rot and mold. Try to get access to the subfloor from below. The rotted pieces will need to be replaced so you could remove them to take a look at the subfloor to try to find how the water is spreading.