Greetings! I've decided to dip toes into HDRI. I modeled my object, and placed it in one of the Cararra preset scenes (the one with the coronet). Here's that render: http://home.earthlink.net/~tacallender/HDRI_case_preset.jpg

Then I decided to get cute, and found a kitchen interior HDRI file, and tried that out. Here's the result:
http://home.earthlink.net/~tacallender/HDRI_case.jpg

As you can see, in the second example, it looks like a gang of five-year-olds left fingerprints all over the model.

I removed all bump settings from my shaders, there are no mixed colors - really, the shaders are quite basic. Does anybody have any idea what might be causing the fingerprint effect?

 There is no one "right" answer for this - it is very scene and HDRI map dependent.  This fix is playing with the following options. The are interelated so while only one may fix the issue, you may need to use a combination of them:

1. increase Photon Accuracy
2.  Increase Photon count
3. Enable "Improved Edges"
4. Engage "Full Ray Tracing"

While this can occur in with all HDRI maps, the low resolution maps  are more prone to creating this problem. If you think of every pixel in the HDRI map as a light bulb, the maps with greater resolution have more "light bulbs" to illuminate your scene accurately. 

Thanks, Mark. I think I had all that stuff toward warp 10, but I'll go back and check.  :)

 If things are maxed out, you may be the victim of low res HDRI maps.