The repair guy discovered that the gas igniter had failed, and the motor and bearings were giving out (explaining that it sometimes ran pretty loudly when first starting up). We decided to replace everything; he replaced the igniter but said he'd need to order a part for the motor bearing.
The next day, the motor quit completely, so the furnace got hot enough to give off hot-metal odors and the house didn't get any heat. The repairman came back and put the new motor in (he'd intended to wait for the bearing part so the whole package could be done at once, but the situation couldn't be allowed to stand like that).
So far, so good.
Thursday morning soon after I went to work, starmalachite got a call that the repairman was coming over "for your appointment". She explained that she hadn't heard of any appointment, and that we were waiting for a part to arrive for the final repair visit. The person at the other end of the phone said that was taken care of. starmalachite asked the dispatcher to tell the repairman come in about half an hour so she could get ready.
After getting ready, and then waiting for an hour and a half, she called to ask what was going on. The answer: "The part hasn't come in yet".
It turned out that the repairman showed up late that afternoon with the part and finished the job. And so, several hundered dollars and thoroughly disrupted day for starmalachite later, we have the furnace back in shape.
I hope that the outfit's repair abilities are better than their organizational abilities. (On second thought, I can chalk this up as a confirmed fact. If the former were as bad as the latter, the furnace surely would have exploded the next time it turned on.)