Only one user in this system has this problem. Every other user is able to load their contacts. When using Outlook or an ActiveSync device, it works fine, but on OWA, it says "Your request can't be completed right now. Please try again later." I did find that it works fine under the light version (using the layout=light variable).
One factor to this might be the fact that the mailbox has existed since 2005. It has been migrated from Exchange 2003, to 2007, and now to 2013. I suspect that recreating the mailbox and moving its content will correct it, but I'm hoping for something less invasive and less impacting on the user.