Can I submit more than 3 reference letters with my grad school applications?
No!

US institutions expect that you have never seen your recommendation letters; your references are supposed to upload/submit them directly to your target departments.  But presumably you are uploading/submitting your own supporting documents.

If the application web site somehow restricts you to naming only three references, then you only get three letters; choose wisely.  If it's a paper application, just ask all your references to mail in their letters.

(I assume you are applying to a graduate program.)