Summary: Fix integers \(n\), \(x\), \(k\) such that \(n\geq 3\), \(k>0\), \(x\geq 4\), \((n,x)\neq (3,4)\) and \(k(n+1)<(n^{n+x})\). Here we prove that the order \(x\) Veronese embedding of \(\mathbb{P}^n\) is not weakly \((k-1)\)-defective, i.e. for a general \(S\subset\mathbb{P}^n\) such that \(\#(S)=k+1\) the projective space \(|I_{2S}(x)|\) of all degree \(t\) hypersurfaces of \(\mathbb{P}^n\) singular at each point of \(S\) has dimension \((n^{n+x})-1-k(n+1)\) (proved by Alexander and Hirschowitz) and a general \(F\in |I_{2S}(x)|\) has an ordinary double point at each \(P\in S\) and \(\text{Sing}(F)=S\).