Figured it out, with the help of Pro Support at Dell. This is a T7910 Precision Workstation which I just changed the mother board on. After trying everything I could think of, compatibility mode, privacy settings, I called Dell where I was smart enough to have bought a 24/7 support contract when I bought the new computer.
After checking a few things, the tech asked if the time was correct on my computer. The clock? I thought, let me check. The gadget clock said 9:50, which was correct, but the clock in the Taskbar said 9:50 AM! I changed it back to PM. The tech said check the BIOS. Not only did was the BIOS time an hour slow, but the date was the 25th rather than the 24th.
I corrected the BIOS, rebooted...and logged right into POTN in IE 11.
Umbelievable. But the tech told me the story of sending out a new motherboard to another client, and the new board would not work at all. After checking some things, they found that the time was incorrect in the BIOS. Correcting it got the motherboard working right away. He said that my being able to log into some sites but not others was a classic example of the problem.
Thank goodness for Dell Tech Support. These guys and women are based in Texas, and there has never been a problem that they couldn't solve for me over the past 15 years.