I have created a simple data acquisition module for measuring 5V signals. There are 18 analog inputs on this board, all of which are a copy of the schematic below.
Each input has an op-amp (MCP604) with unity gain. This op-amp should work to within 60mV of VCC, but in half of my circuits, it won't get any higher than 4.3V. (Notice the 100K pull-up to 5V, the input pin disconnected, should output very close to 5V.) In the other half, the circuits operate just fine. I have two of these boards built, and most (not all) of those circuits are limited to 4.3V. Under 4.3V, the circuit will behave in a linear fashion.
It doesn't appear to be a glitch at power-up, the circuits that work always work, the circuits that don't always don't.
I've cut the trace at the output of the amplifier to see if the A/D converter was clamping it somehow. No effect.
I've tried adding load to the output, with different values of R50, 10K to infinity. No noticeable effect, maybe a couple of millivolts.
Probably an unnecessary detail, but for the sake of completeness... I need to get the zero input of the signal within the linear range of the amplifier, so I boost it slightly with the 20K/100K voltage divider. This lifts a 0V signal to 0.833V. The device connected to this circuit may provide an input voltage of slightly less than 0V (sometimes a low as -0.5V).