Just so it's clear, these were individual labs at universities that rejected his request. There are a lot of legitimate reasons they may not have wanted to take him on, including:

-Not enough physical space in lab - most wet labs are crowded with 2-3 people per bench

-No one with enough time to supervise a high school student bumbling around the lab. You basically need a grad student or post-doc babysitting new people (HS student or new grad, doesn't matter) since they don't know where anything is or how to do anything

-Not enough money. Funding has to come from somewhere, and if they don't have money to throw at this particular project they won't.

-Not the focus of the lab. I would hazard a guess that the HS student emailed anyone that was remotely involved in pancreatic cancer or material science. Realistically, there are probably a handful of labs that do this sort of research and the kid just didn't know which ones to email...so he emailed everyone.

-Other. Maybe the professor is teaching? On sabbatical? Already has ten undergrads?

The lab he worked in seems very well-suited for this project. I find it unlikely that there are really 200 labs in the Baltimore/Washington area that are. So I'm guessing much of the reaction is "good for this kid, it never would have worked here."