Thanks for the idea, tried this out with my HP 8562A. The effect is quite clear.
With the noise floor as the slightly lower-intensity saved trace, sweeping the full lower band, 0-2.9GHz, ref level 0dbm, everything else "auto":
First picture I input 1.95535 GHz at 0dBm .
Second picture increase to 10 dBm. YOW!
Third picture 1GHz at 10 dBm.
Fourth picture 2GHz at 10 dBm.
Now, I suppose if you are dealing with a controlled situation and you want to make sure this situation doesn't mess up your measurements, you could increase the input attenuation to where linearity is restored, look at the full span, determine that there is this undesired interference, add a filter, etc.
But if you wanted to make a (quite expensive, but self-contained) instrument to cope with this condition without having to go for a much narrower bandwidth frontend architecture, would the most sensible thing to do be to just have two (or three!) separate frontends with different first IF frequencies and switch between them as needed (or automatically with software)?