Let $V$ be a variety (or a Whitney stratified space); $C$ is a constant etale ('topological') sheaf on it. Let $t$ denote the middle perverse t-structure for the corresponding derived category (of sheaves) $D(V)$. My question is: how to find an $e$ as large as possible so that $C\in D^{t\ge e}$? The problem is that for a singular $V$ and an embedding $i:T\to V$ it seems difficult to compute $i^!C$ (as well as the Verdier dual of $C$). 


In particular, I would like to obtain an interesting estimate for $e$ in the case when $V$ is smooth in 'large' codimension (say, that is not much smaller than the dimension of $V$), but whose singularities are 'very bad'. 



How can one bound 'the lower perverse degree' for a constant sheaf on a variety that is smooth in high codimension?
Let $n$ be the dimension of $V$. I assume that by "constant sheaf", you meant "constant sheaf shifted in degree $-n$", so that if $V$ is smooth, $C$ is indeed perverse.


Let us call $e$ the best possible integer as in your question. Then we generally have $e\geq -n$, and if the variety is smooth we have $e=0$. If I understand your question, you would like an estimate that tells you that, if the singular locus has "big" codimension, then e is close to $0$.


However, let me try to build an example of a $n$-dimensional variety with a $0$-dimensional singular locus for which $e=2-n$. If it works, then it rules out the possibility of any interesting estimate as you wish.


Start with a smooth $n-1$-dimensional subvariety $X$ in projective space $P^{N}$ and let $V$ be the associated cone in $A^{N+1}$. The singular locus is the origin $o$. Let $U$ be the open smooth complement of $o$ and let $i:o\hookrightarrow V$ and $j: U\hookrightarrow V$ be the corresponding immersions.


There is a distinguished triangle 
$$ i^! C \longrightarrow i^\* C\longrightarrow i^\* Rj\_\* C \longrightarrow $$
and an isomorphism
$$ i^\* Rj\_\* C \simeq R\Gamma(U,C) .$$
The latter comes from the $\mathbb G\_m$ action on $V$ (see Lemma 4.5 in Kazhdan-Lusztig paper "Schubert varieties and Poincaré duality").


Taking $C=\mathbb Q\_\ell[n]$ , we get $H^{2-n}(i^!C)\simeq H^1(U,\mathbb Q\_\ell)$. So it remains to find $X$ such that $H^1(U)\neq 0$. 


But $U$ is a $\mathbb G\_m$-torsor over $X$ and thus is the complement of the zero section of some line bundle on $X$. Therefore we have a triangle
$$ R\Gamma(X)[-2] \longrightarrow R\Gamma(X) \longrightarrow R\Gamma(U) \longrightarrow $$
where the first map is the Lefschetz operator on $X$ associated to $U$.


This yields an isomorphism $H^1(X)\simeq H^1(U)$, so it remains to take $X$ with $H^1(X)\neq 0$. 


Hope I didn't write too much nonsense.


