Is It Possible for an Invisalign Button to Break but Not Fall off Completely?

Short answer to your question is yes, part of an Invisalign button aka attachment can break off, in which case it should be repaired. However, many attachments in use today do not fill the aligner form fully, to allow for concentrated force delivery, and room for movement. Your doctor can tell you what the situation is in your case.

It’s possible for an attachment to wear out or partially break of over time. Depending how far along you are in treatment your doctor might choose to not replace it. If the tray is not sitting in all the way that might also cause the attachment not fully fill in the mold.

It is indeed possible for a button to partially break.  If you are midway or partway through your treatment and need a button replaced, your doctor may have to order an attachment template from invisalign that correspondes to the tray number you are now using.  This can take a couple of weeks.