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Greetings from the Truman Library,
Thank you for your question! I would say that many, but perhaps not all, letters to the President are saved and archived. In some cases, for high profile topics on which the White House received a lot of mail, the White House File Room might have only kept a sampling of the correspondence they received. In addition, if the letter contained a question that was better answered by another Executive Department, such as the State Department or Justice Department, the actual letter may have been forwarded to that office, and only a referral note would have been kept in the White House Files. While these materials are searchable, the vast majority of them are not digitized, and each Presidential Library is a little bit different in terms of the organization of the White House Files in their collection. Additionally, due to the COVID-19 public health emergency, Library staff are mostly working from home and do not have access to their physical location to search for documents. It would be helpful for Library staff to have as much information as you can provide about what you're looking for, such as the name of the person, a more precise date, and possibly the topic or subject of the letter.
If there is anything else I can do for you, please let me know.
To piggyback off of Tammy's answer, the newer the library, the more likely it is that only a sample of correspondence was kept by the White House. This is because as technology has changed (and the population of the U.S.), the White House receives much more correspondence (in terms of volume). Since you are looking for correspondence from the 1950's-1960's, there's a higher probability that the bulk of presidential mail was saved. As Tammy said, you will need to contact the libraries that cover those time periods and provide as much information as possible to help them identify records responsive to your request.