Let $X$ be the blowing-up of $\mathbb{P}^n$ in $r$ points in general position. Let $\{H,E_1,...,E_r\}$ be the standard basis of $Pic(X)$. Further let $D=dH-\sum m_j E_j$, be an effective divisor, with $m_j \in \mathbb{Z}$. Is it true that if $m_1\geq 0$ then $E_1 \not\subseteq Bs(|D|)$?

If we suppose all the $m_i$'s are non negative, this corresponds to say that $dim |dH-m_1E_i...-m_rE_r|>dim |dH-(m_1+1)E_1-...-m_rE_r|$. In other words this means that if we consider all the hypersurfaces of $\mathbb{P}^n$ of degree $d$ passing through $r$ general points $\{p_1,...,p_r\}$ with multiplicities $m_1,...,m_r$, then they do not all pass through $p_1$ with multiplicity $m_1+1$. My intution brings me to say this is obvious but I cannot prove it. In fact everything is trivial for nonspecial linear systems, but, a priori, special linear systems may be a problem.

Note that the generality of the points is necessary: for example you can simply take $\mathbb{P}^2$ blown up in 3 collinear points and note that $E_1 \subseteq Bs(|H-E_2-E_3|)$.