Calendar invitations not accepting responses on iPhone and MacBook

I've used iCalendar for YEARS. Recently any invitations that I receive, even when I click a response, pop back up in the NEW invite area and I can't permanently get them to stay accepted (or declined, or "maybed"). This is happening both on my iPhone and my MacBook Air. Please help...

[Re-Titled by Moderator]

Original Title: Calendar invites can't be accepted, even after clicking decline, maybe, or accept - both iPhone and MacBook