Stratix III JTAG Failure

First I would measure that 1K resistor to ground and make sure it's really 1K. If that's not the problem it really sounds like you've got a solder short somewhere. There are tools out there that can pinpoint where a short is (simply by measuring resistance to a very large number of significant digits). For example, try measuring the resistance to ground both at the JTAG header and at the via where it connects to the FPGA.

What is your relationship like with the assembly house? We often have the assembly house x-ray the board when something like this happens. If this is happening consistently, it points to an assembly problem (too much solder paste, too much solder).

It could possibly be a layout problem. What tool did you use for layout? Is there anywhere in the layout where you could see a danger of TCK shorting to ground.

Is this a lead-free board?

Have you done any rework on the board? I've often seen people leave flux on the board after doing rework, then they power up the board and it works for a while until the solder heats up under a BGA and the flux residue makes it flow.

Jake