Oh dear girl!! Where were you when we were trying to figure out the gifting system? You've been here long enough.
You can only receive 20 responses each day. Sending out more than 20 only causes the gift request system to jam up for some reason. It seems that when this happens some of the requests do not go out at all and when you requests more gifts the next day the problem worsens and everything grinds to a halt. This accounts for those times when we see requests from neighbours that are days old.
So. What to do? Do not send out any requests for about a day or even two. Not even one request. Use this time to go through your neighbour list and find out who is still active and who is not. This should be easy enough. Delete any inactive neighbours and you could also ask neighbours to send you the required part. Ask them on their walls. Some will, some won't. But at least you will get some parts this way. You should also see those parts start to come in from your old requests as the system unclogs itself and sends them out to your neighbours.
After a day or two send out ONLY 20 requests for parts for whichever appliance you are building. Only 20. Then the next day send out another 20 requests. You can try to figure out who responds to your requests and keep sending them requests each day.
This should solve your problem and you should find that your neighbours are more helpful than you think.