I have a similar issue, but not identical to the others listed. I have an AT&T Galaxy S5. I use the gmail app rather than the samsung native email application (so I dont have the same messages going to 2 different programs on the same phone) I also have thunderbird on my laptop set to Not delete messages from Server. My problem is, when my computer is off, and I am reading my emails on the phone all night, when I try to access them from thunderbird the next day they are nowhere to be found. I have tried switching the settings around a couple times, but it only seems to happen if the computer is off for a while. If I have thunderbird closed, read on email on my phone via the gmail app, then open thunderbird, they are there. If I have the computer off, read emails on my gmail app at 10pm, then log into my computer 12 hours later, they are not there.