With Curve25519, by contrast, DJB explains exactly what constraints were imposed (with very solid justifications for each of them) and then proves that Curve25519 is the unique solution to these constraints which minimizes the remaining free coefficient (which maximizes efficiency). NIST should appoint him to be their Czar or something.
NIST P-256 (and ECDSA in general, to some extent) is widely suspected to have been subverted by NSA, as outlined in the first link in octoberfranklin's response, page 16 of [1], and [2].
P-256 is not known to or even suspected to have any backdoors.