I've been doing $3k per month last 6 months through mobile deposit with USAA all with Publix WU money orders, no problems so far. Anyone with issues might have to do with the age of your relationship and the amount of money you move through it on a regular basis and your deposit limits.
My daily deposit limit is $100k for instance. It was $5k for a while, then $10k and jumped to $100k in the last year. I've been with them for almost 15 years and I usually have $5k to $15k in my checking account at any one time. I've had investment accounts with them but dumped them for Vanguard. I used to use their insurance, but it jumped up to be more expensive than most anyone else even with a perfect driving record and being with them for almost 10 years, so I dropped them for whoever is cheapest. I've also had their rewards card for over 12 years although I only use it anymore to make sure it stays open.
So basically, it's YMMV. Take it slow at first and then ratchet up and if you get a call, just be honest and do what they ask of you. I used to value their relationship more since they were cheaper on insurance and were one of the first banks to allow deposits from a scanner then from a phone, but that's standard now and their insurance is expensive, so I don't really care if they shut me down although I'm not pushing it.