After many frustrating experiments/failures, I found these setting working great on Consumer Cellular network with my Moto X 1st (2013):
Name - Consumer Cellular [you can name anything you like, really)
APN - ccdata [I personally find "att.mvno" just does not work at all]
Proxy - Not set
Port - Not set
Username - not set
Password - not set
Server - Not Set
MMSC - [Look at the attached pic]
MMS proxy - [Look at the attached pic]
MMS port - 80
MCC - 310
MNC - 410
Authentication type - None
APN type - default,mms,supl,fota,hipri,agps
APN protocol - IPv4
APN roaming protocol - IPv4
APN enable/disable - APN enabled [on my phone, this option is greyed out]
Bearer - unspecified
MVNO type - GID [on my phone, this option is greyed out]
MVNO value - 2ac9 [on my phone, this option is greyed out]
*How to proceed:
1. Power off and then power on your phone, a fresh start.
2. Make sure "cellular data" is switched on, "Restrict background data" is off, Wifi is off [Wifi can be switched on after you confirm that the settings work]
3. Go to APNs section and create a new set of APN settings as above, then save it. You should notice the signal strength icon changing after saving.
4. Waiting 15 seconds then begin the first test: sending a MMS picture message to yourself, namely send to your own number. For the second test, go to your email app and send another picture to yourself (make sure the pic is less than 1MB in size, the smaller the faster). Since CC network is AT&T network, the email format is yourcellphonenumber[at]mms.att.net [for example:1234567890[at]mms.att.net . Third test, reply to your lastest message, namely your own email, with a MMS picture message. They all should go through. Keep the picture size down and allow 60 seconds for each test.
*In my experience, powering your phone off (then on again) can cause MMS problems again. I have not figured out the reason. So if all the tests go through successfully, DON'T POWER your phone OFF.
If you have to do so, you will have to go back to your APNs screen and modify the above settings a little. For me, I removed "hipri" from APN type, then performed the three tests. They were all successful, of course. Then I went back to my APN setting and added "hipri" back. And then I performed the three tests again. The point is I revive my APN setting by changing it a little because after changing something is reset.
Extra information for tinkers who want to troubleshoot further:
Phone model: AT&T Branded Moto X (XT1058); SIM and Bootloader Locked; Android 5.1
My location: San Francisco Bay Area, California (Will this settings work in another geographical area? I don't know. You tell me
