Tillman was in business from 1859 until 1962....your desk appears to be a style that was popular about 1910....with that pull out writing surface.
The chair is great.....probably from the same era or a bit newer. You can make the finish look good with something like Formby's finish restorer....and the rush seat can be repaired.
But I would spruce up the finish and put a cushion on the seat and use it....no reason why it shouldn't be sat in.
Linda C

Thanks, Linda. I always appreciate your knowledge. The desk writing surface actually folds down. As for the chair, I had taken it to an old guy who was an "expert" in rush seating. He told he it couldn't not be repaired....maybe he just didn't want to do it. I'll try to spruce these things up.

Maybe he meant couldn't be repaired without re-doing?
What supports the writing surface? Does it have pull-out supports?

No supports, you have to pull out the drawer and it rests on that.

On the chair, he said the rush was put on while the chair was being constructed, and thus the chair would have to be taken apart to put new rush. I suppose it all make sense.

That type of chair always has to be taken apart to put on new rush!! I expect your "expert" was not.

Here is a link that might be useful: Re rushing a seat

from the link: ...cost, somewhere between $135 - $285. to redo the seat. Maybe your expert meant it isn't practical to put that much money into a $25 chair.

A splat back Windsor style chair with a new rush seat is not a $25 chair.