OK. I found a solution.
I tracked the issue to modsecurity and a MULTIPART_UNMATCHED_BOUNDARY message. In the /etc/modsecurity/modsecurity.conf file, I commented out these two lines
SecRule MULTIPART_UNMATCHED_BOUNDARY "!@eq 0" \
"id:'200003',phase:2,t:none,log,deny,msg:'Multipart parser detected a possible unmatched boundary.'"then restarted apache. I no longer have upload issues. (Although I'm not sure what the implications are for the security of the site...)