We prove blow up in finite time of the solutions to some reaction-diffusion systems preserving nonnegativity and for which the total mass of the components is uniformly bounded (these are natural properties in applications). This is done by exhibiting explicit counterexamples constructed with the help of a formal computation software. Several partial results of global existence had been obtained before in the literature. Our counterexamples a posteriori explain why extra conditions were needed. Negative results are also provided as a by-product for linear parabolic equations in non divergence form and with discontinuous coefficients and for nonlinear Hamilton-Jacobi evolution equations.