Hi Martin,
There are a few things going on here.
First of all, I checked this in our lab and my input current measurements agree with yours. I'll post my graph below for your reference. This is not very surprising considering the conditions where the AD8421 input bias current is specified are at ±15V supplies and an input voltage at 0V. The data sheet shows the ±15V behavior for bias current vs common-mode voltage in figure 18. It is clear in figure 18 that the bias current starts to knee up as the input stage begins to saturate, which is slightly before the input actually limits. This is common amplifier behavior. For a single +5V supply, there is simply much less room for the input voltage to swing before this happens. Here is what I observed (input range limits are marked with a dotted line):
Secondly, Figure 13 in the AD8421 data sheet is wrong. The scale got altered while the figure was being illustrated and as a result it looked like you can go outside the input range. This has been reported to us already and we are working to get it fixed.
Third, the web tool should not have been saying that the AD8421 was out of range in the configuration linked by Achim. I have contacted an engineer in the web tools department about this. She has already found the problem and is working to fix it. The graph shown in the tool is correct. There was a bug in the function that checks if you are outside the limits and generates the error message.
Back to your circuit, if there is not an easy fix by adjusting the op amp gain, filter values, and/or supply voltage, we can work on recommending another solution. It looks like you chose your amplifiers for low wide-band noise. Is that correct? AD8656 is great for higher frequency low-noise measurements, but is a straight CMOS op amp, so near dc, the 1/f noise is probably higher than you bargained for.
I think the best overall solution in terms of noise/power/space would be to build a differential gain stage with ADA4528-2 (which is zero drift, so it has no 1/f noise). The next stage could be something like the AD8476 low power differential amplifier/ADC driver to provide high CMRR and the differential outputs required by the ADC. Feel free to contact me directly and we can discuss details: scott.hunt at analog dot com.
Sorry for all of the confusion. I hope this helps.
Best regards,
Scott