I enter my comment as an answer. The paper *On the normal bundle of submanifolds of $\mathbb{P}^n$* by Lucian Badescu contains some answers to the question. Here are the links:

Published version: http://www.ams.org/journals/proc/2008-136-05/S0002-9939-08-09255-1/

On arXiv version: http://arxiv.org/pdf/math/0701487v1.pdf

In particular, Theorem 1.2 (due to Faltings) in the above reference is of interest, in connection to this question.