I have access to the servers. I checked SharePoint logs, but didn't see any obvious error, everything looks Medium when this happens. IIS logs have lots of HTTP status codes, like 200, 207, 401, 405, 414 for the site and the web service I'm using. Can't tell which one is relevant. I know 401 is access denied, but it's there for many things.
I was googling for possible claims issues with InfoPath and I found a few articles, like below, talking about some configuration needed to make the web services work in InfoPath for Claims web applications. Do you still think it's not Claims issue?
http://www.c-sharpcorner.com/UploadFile/Roji.Joy/how-to-enable-sso-for-web-services-used-in-infopath-form-in/
http://blogs.technet.com/b/rajbugga/archive/2013/08/07/infopath-over-claims-authentication-sharepoint-2010-amp-2013.aspx
http://suehernandez.wordpress.com/2013/10/11/sharepoint-2013-infopath-claims-getuserprofilebyname/
Thanks,