The Pandora's Box problem and its extensions capture optimization problems with stochastic input where the algorithm can obtain instantiations of input random variables at some cost. To our knowledge, all previous work on this class of problems assumes that different random variables in the input are distributed independently. As such it does not capture many real-world settings. In this paper, we provide the first approximation algorithms for Pandora's Box-type problems with correlations. We assume that the algorithm has access to samples drawn from the joint distribution on input. Algorithms for these problems must determine an order in which to probe random variables, as well as when to stop and return the best solution found so far. In general, an optimal algorithm may make both decisions adaptively based on instantiations observed previously. Such fully adaptive (FA) strategies cannot be efficiently approximated to within any sublinear factor with sample access. We therefore focus on the simpler objective of approximating partially adaptive (PA) strategies that probe random variables in a fixed predetermined order but decide when to stop based on the instantiations observed. We consider a number of different feasibility constraints and provide simple PA strategies that are approximately optimal with respect to the best PA strategy for each case. All of our algorithms have polynomial sample complexity. We further show that our results are tight within constant factors: better factors cannot be achieved even using the full power of FA strategies.