I have two HP-19C's that have the exact same strange behavior. I've had trouble with the batteries on both, or so I thought. I will get them working, then a few days later, they don't turn on. I didn't realize until today that I didn't need to "fiddle around" with the battery packs. After a few minutes of removing them, adjusting the cells, etc., they would finally turn on.
Today, I just spent time turning the power switch on and off. After a few minutes, and some intermittent powering up, they both power up reliably.
Any idea what component has begun to fail in both? I assume it's a capacitor. Has anyone else had observed this strange behavior?