Two parts. If the form is anisotropic, there is a specific prime $p$ for which the form is anisotropic. The problem is that the form does not integrally represent anything in the $p$-adic squareclass of the discriminant of the form. This follows from just a few pages in Cassels, rational Quadratic Forms, which I recommend; mostly pages 58-59.
Next, it is possible to have exceptions that do not directly arise from congruences. The famous example of Siegel is the genus with two forms, $$ x^2 - 2 y^2 + 64 z^2, $$ $$ (2x+z)^2 - 2 y^2 + 16 z^2. $$ Note that the binary $4x^2 + 4 xz + 17 z^2$ is indeed in the same genus as $x^2 + 64 y^2.$