That notion, presented in a recent study, is ridiculous, as Roger Clegg lays out over at Minding the Campus:

The results are well-summarized in the study’s graph (figure 1). Basically, it shows (a) that blacks and whites agree that discrimination against blacks has gone dramatically down over the last half century, and that discrimination against whites has gone up; but (b) that blacks believe there is more antiblack discrimination still than whites do; and, finally, (c) that whites believe that there is now actually more antiwhite discrimination than antiblack discrimination, while blacks think that the amount of antiwhite discrimination is still negligible. . . .
. . . The only remarkable thing about the study is not the data, but the extremely dubious conclusion the researchers draw from them, namely that they show somehow “that whites see racism in zero-sum terms”: that if you have less antiblack bias then there is going to be more antiwhite bias, and so progress on civil rights is a threat to white people. There is nothing in the study to support the notion that whites see the two historical trends as linked in this or in any other particular way, and the obvious reason that whites (and, for that matter, blacks, albeit to a lesser extent) see more discrimination against whites now and less discrimination against blacks is because, well, there is more discrimination against whites now and less discrimination against blacks.