[rjknight]

No, but Drupal's out-of-the-box page cache caches pages in... the database.

Even with pages cached in Memcache, Drupal would open a DB connection by default (it's possible to override this using page_cache_without_database, but not everyone is going to do that).

Using something like Varnish as the primary page caching mechanism would work around this, albeit with some trade-offs.

Generally though, the fault with issues like this is often that the MySQL connection limit is set lower than it needs to be. Often, this is set at a level far below that which would actually cause the DB server to become overloaded, and it just starts rejecting further connections even if capacity exists to handle more.

EDIT: typos