Mori Dream Spaces were introduced by Hu and Keel as normal, $\mathbb{Q}$-factorial projective varieties whose effective cone admits a nice decomposition. Mori's minimal model program can be run for every divisor on a Mori Dream Space. We study the question whether the blow-up of a weighted projective space at a general point is a Mori Dream Space. For every $n\geq 3$, we find a sufficient condition for the blow-up of $\mathbb{P}(a,b,c,d_1,\cdots,d_{n-2})$ at the identity point not to be a Mori Dream Space. We exhibit several infinite sequences of weights satisfying this condition in all dimensions $n\geq 3$.