I think the prices for most of these journals individually is enormous - they are usually sold in bundles through individually negotiated, confidential pricing agreements that run into the hundreds of thousands of dollars. Far cheaper per journal, but probably including a bunch or journals that the institution normally wouldn't be interested in, and a ton of garbage.

http://www.pnas.org/content/111/26/9425.full

Not to say that many/most papers can't be found with a little bit of effort, and I can't really imagine a situation where an establishment journalist emails authors about one of their papers and doesn't get it.