The LightSquared mess is partially due to a change the FCC made to allow ground augmentation for satellite services (note that XM-Sirius also heavily utilize this rule change). LightSquared is using it to put in a nation-wide mobile broadband service in the L band, 1525-1559 MHz. GPS L1 sits at 1575MHz.
The problem lies in the receivers, in particular commercial and consumer GPS receivers will not be able to tolerate the interference and will end up with overloaded front ends. The military units might be OK due to their anti-jamming tolerance (this is more or less broadcast jamming) and future consumer designs might be able to tolerate it (at greater cost). Both are presumptions though, filtering might end up not being practical.
To put some numbers to it, a terrestrial GPS receiver at sea level sees about a -160dBW signal, which is below S1 on a typical HF radio and almost at the receiver noise figure of even the best HF radios (this is the incident, natural noise of the electronics of your radio with the RX antenna grounded). The LightSquared terrestrial stations will be I understand as much as 15kW, which if true would represent somewhere around +40dBW in the near field. Your radio at S9 is -105dBW, so that would be called 'S9+145dB'. Not only will it kill any reception, it's probably going to just blow the receivers out. Another analogy, your GPS signal is like trying to see a LED flashlight next to a Coors Field flood light.
It's really LightSquared's problem as they are going to be spattering and are the interfering party. But with tacit FCC approval. It's not that the spectrum must be shared, it's that the ground stations are going to essentially be GPS black holes and are being pursued even though the FCC technical office and most of the industry knows this will likely end up in problems.
Also realize that this is primarily going to be a problem only close to the ground stations, of course LightSquared is planning 40,000 of them. None-the-less, if you're a few miles away it should not be a major consumer problem. The ones of concern would be permanent or semi-permanent ones located close, such as the FAA WAAS systems for example.