iPad personal hotspot turns itself on and off

I have an iPad Mini + cellular. My wife has an iPad 2 + cellular. Both are running iOS 7.0.6 (after reports of 7.1 maybe decreasing battery life, I’m waiting for 7.1.x).

Recently they both have developed the same problem. If you turned on the hotspot, neither the MacBook, the MacBook Pro, nor the Apple TV could “see” it. Going through the whole rigmarole of resetting everything under settings didn’t help. Occasionally, rebooting the iPad in question would temporarily solve the problem, but not always.

Because of where we live, our cellular network is the only available internet, so this problem is really annoying.

We read somewhere about Apple collaborating more closely with the fat telecoms to prevent “illegal usage” of hotspots. (Com’on, if you’re paying for data you’re paying for data and should be able to use it! grumble, grumble) So we figured out to go to Settings / Cellular Data / APN Settings, and scroll down to “Personal Hotspot,” click “Rest Settings” / “Yes” (or “Okay”), and fill in the extra login info from our provider.

Then we turn off cellular data and the personal hotspot, wait a few seconds, and turn them back on. That works. Sort of. Sometimes.

But it doesn’t work for very long. Sooner or later, the connected device thinks it's still connected, but the iPad is no longer sharing connectivity beyond itself.

It does not matter if bluetooth is on and wireless is off, or if bluetooth is off and wireless in on, or if they are both on, or even if they are both off and a computer is tethered with a USB cable. We’ve tried every possible combination.

It also doesn’t matter if we keep the iPad awake or keep the personal hotspot settings page open on top.

When either iPad is acting up like this, I’ve noticed that the blue notification bar at the top of the screen telling whether anyone is connected comes and goes. Even when it’s not there, a connected Mac thinks it’s still connected, but there’s no functional connectivity.

Any help?