can $E_c(T)=\{x\in X~:~\nu(T,x)\geq c\}$ have isolated point? where T is a positive current of bidegree 1, c is a positive real number, $X$ is a complex variety, and $\nu(T,x)$ is the Lelong number of T on x.

Yes, it can. For example take $X=\mathbb{C}$ and $T=dd^c\log z^2$ (up to a positive factor this is the delta function at the origin), then $T$ has positive Lelong number at the origin (with the correct normalization the Lelong number is $1$) and zero outside. 

