Sorry to respond to your question with a set of questions, but there are some things to consider because the old press releases may provide value in ways that might not be obvious. I'd ask yourself or the client these things:
1. Public companies are required to report and archive company news and financial information. If its a public company, then I'd keep them because press releases are a way to satisfy this legal requirement.
2. Is there even a small chance that the releases would need to be cited for something? Say a customer or reporter asks a question (over e-mail or on social media or something) -- it could be answered by sending a link to an old press release. Plus, old information can always be valuable for "institutional knowledge" -- say a bunch of longtime employees leave, then new hires could read them to get up to speed on what the company has done in the past.
3. Might the press releases contain text that could be found in relevant, long-tail searches? If so, I'd keep them.
4. Are there external links to the press releases (check Open Site Explorer or similar tools)? If so, I'd keep them (and obviously 301 redirect the links to the new site).
These are just a few issues off the top of my head. In general, I think that there's never a case when a website can have too much information or content (as long as it's managed well). Unless you're talking about hundreds of pages, I don't see how a migration would be too much of a pain. Plus, it's always good to have extensive company archives because, well, you never know!
I can tell you that the old press releases on my company's site get a bit of traffic -- both straight from organic search and from people navigating there from other pages. I'm sure the same is true for most sites.
Again, this is just my two cents. Whatever you end up doing, good luck!