No, to answer a number of the questions I've received on Twitter in the past week, the New York Giants do not have the necessary cap room to acquire Houston's Andre Johnson. (I also don't think the Texans are trading him, but whatever.) But they do have enough to sign their draft picks, and in a couple of weeks they will have a fairly sizable amount.
NFLPA records show the Giants with $3,900,091 in salary-cap space at this point -- more than only six other NFL teams. We don't know for sure how much they're allotted for their rookie pool, but it's likely to be something in the neighborhood of $2.5 million. So that leaves them with about $1.4 million right now if they want to go out and sign a free agent. And yes, tight end Jermichael Finley is still out there, though the medical reasons for not signing him likely outweigh any financial ones.
It's also important to note that the Giants will add another $5 million in cap space on June 2 -- the result of designating center David Baas a post-June 1 release.
So as of June 2, the Giants should be around $6.4 million under the cap even with all of their draft picks signed. It's hard to see how they could use all of that this year, but they do retain the option to roll any unused cap room over into 2015. The Giants were one of only three teams (St. Louis and Washington were the others) that didn't roll over any extra cap room from 2013 into 2014.