I agree with Epson.. the problem is Yosemite and it is bug central at least on the network side.
The problem is not the airport router as much as the OS in the computer.. as is clear by the fact it still works fine with NOT Yosemite.
However to get Yosemite working it is sometimes necessary to start again on the airport setup. So.. here we go.
Factory reset universal
Power off the AE.. ie pull the power cord or power off at the wall.. wait 10sec.. hold in the reset button.. be gentle.. power on again still holding in reset.. and keep holding it in for another 10sec. You may need some help as it is hard to both hold in reset and apply power. It will show success by rapidly blinking the front led. Release the reset.. and wait a couple of min for the AE to reset and come back with factory settings. If the front LED doesn’t blink rapidly you missed it and simply try again. The reset is fairly fragile in these.. press it so you feel it just click and no more.. I have seen people bend the lever or even break it. I use a toothpick as tool.
Then redo the setup from the computer with Yosemite.
1. Use very short names.. NOT APPLE RECOMMENDED names. No spaces and pure alphanumerics.
eg AEgen5 and AEwifi for basestation and wireless respectively.
2. Use all passwords that also comply but can be a bit longer. ie 8-20 characters mixed case and numbers.. no non-alphanumerics.
3. Ensure the AE always takes the same IP address.. this is not a problem when the AE is the main router. It is a problem when it is bridged.
4. Check your share name on the computer is not changing.. make sure it also complies with the above.. short no spaces and pure alphanumeric.. Yosemite has a nasty habit of ghosting itself on the network and repeatedly changing its name.
5. Make sure IPv6 is set to link-local only in the computer. For example wireless open the network preferences, wireless and advanced / TCP/IP.. and fix the IPv6. to link-local only. Even in the latest Yosemite, this is a fail when set to auto.
eg for wireless.
No luck we need more info.
What is the main router? Make and model please.
What model is the airport?
How is it connected and used in the network.. ie. bridge or router.. ??
Any issues showing up in the airport utility?