Instrumentation Amp problems
Well, you don't disclose your power supply voltage, but I can guess what is going on. Refer to the figures in the datasheet as this one: 



Your Vref is 0V, Vs = ??V, single supply, and your G = 1000, but you can see that the output voltage cannot safely exceed about 1.4V. That limit is the same with Vs = 5V, so it doesn't matter that much what your supply voltage is.  

Now, if you give it a minus supply, such as +/-5V, things really look a lot better:



The reason for this is that internal nodes inside the chip cannot be allowed to saturate, so even though the inputs and outputs appear to be within range, the instrumentation amplifier won't function properly unless it has enough internal headroom. 

It's unlikely they'll put information that's irrelevant to you in the datasheet (sometimes they do leave things out that you might find important or might not be a 'widely known' as they assume)- it's vital to make sure you understand everything in there.