The Zombie Stack Exchanges That Just Won't Die
For example - books published before the 1950s.
University libraries will continue to purchase more and more books, but there is only so much space that is available in each university libraries. Some libraries (like MIT's) seem to have more modern collections than other libraries (like Brown's) though.
InquilineKea
First, early-to-mid 20th Century books are not usually considered "very old"; very old books are usually stored in special collections.
Many libraries (including both Brown and MIT) store older, low-use materials in offsite, high-density storage facilites. Others, without the budget to do so, may choose to discard these books, particularly if they're widely held by other libraries and do not get used often.
Collectively, libraries are investigating their collections for unique and rare items, to be sure that enough copies exist to keep them available. This is known in the field as the "last-copy problem."
You may be interested in the Ithaka report "What to Withdraw?" which asks when the availability and quality of a digitized version of a print item warrants withdrawal of the item from circulating print collections.