I don't really see a reason for this not to work the way you expect with the connections you have listed here. Might I suggest some debugging steps?
1) See what happens to the output range when you put a large-ish Rg ( say 27k ohms) in the circuit. Does the output swing change at all?
2) Have you checked the inputs to make sure they are as you expect with the amplifier connected? If so, how about using a larger resistance POT to simulate your resistor divider just to remove the sensor from the picture until you get the circuit working.
3) Perhaps the simplest explaination might be that the INA126 itself is bad. Do you have a spare you can drop in to test?
Here is what I was thinking before:
This is a basic inverting opamp circuit. L1 and C2 are only there to filter the power supply to the opamp a little. They are not strictly required, but might reduce noise depending on what else is going on around this circuit and how clean the 3.3 V supply is.
R3 and R4 form a voltage divider that holds the + input at a fixed 500 mV. C1 removes noise that might be coming from the 3.3 V supply.
The real meat of this circuit is R2, the feedback resistor, and R1, the thermistor. The way the feedback is arranged, the opamp will do whatever it takes to keep its - input the same as the 500 mV on its + input. That means there will be a constant 500 mV accross the thermistor. The current thru the thermistor is then a function of its temperature. This current only comes from R2, so the voltage accross R2 is inversely proportional to the resistance of the thermistor.
This circuit won't utilize the full A/D input range, but will do significantly better than just a bare resistive divider as was discussed in your previous question. At 140 Ω, the current will be 3.57 mA, which produces 1.68 V accross R2. This voltage on R2 is added to the 500 mV bias, so the minimum OUT value is 2.18 V. At 98 Ω, OUT is 2.90 V, for a total range of 719 mV. That would be 223 counts of your A/D, which is more than typical thermistor accuracy can support.
You can get a wider output range by using a lower bias voltage and making R2 bigger accordingly. The value of R2 is directly proportional to the gain of this circuit. I showed 500 mV as a example because it seemed like the maximum sufficient value, but 250 mV would give you more than twice the A/D range. I wouldn't go much lower than that since other errors and sources of noise would start to get significant.
One advantage of this circuit is that it keeps a low voltage on the thermistor, which makes self-heating negligeable. At the worst case, the 500 mV is applied to 98 Ω, which causes a dissipation of only 2.6 mW. If you use 250 mV bias it goes down by a factor of 4 to 640 µW. Unless you have a very unusual situation, that amount of self-heating should be irrelevant.
One issue to keep in mind is that the output is dependent on the 3.3 V supply level. However, since you specifically mentioned 3.3 V, it sounds like it is produced by a regulator, so should be fine.
If you only have 3.3 V power available, you need a rail-to-rail opamp as I show. A TL081, for example, would not work here without both higher and lower supply voltages.
Best Answer
It's amplifying the voltage across the RTD. The 10k resistors in conjunction with the 100nF caps have a filter time constant of 1ms.
The common mode of the instrumentation amplifier would not support low voltages when at 5V as shown in the graphs below (at a gain of 10 the common mode would be between the blue squares as only gains of 1 and 100 are shown). If the output of the amplifier were at 5V then the input needs to be centered around 2.5V or the instrumentation amplifier would not gain up the signal correctly.
The ability to amplify all of the RTD's resistance range.