Half my power out, but firing up an appliance turns it back on
Call the power company back. You lost a pole.

Here's how your house normally works.



Here's what happens when a hot wire breaks. 



That's simple enough, right?  Half the 120V loads croak.  And all the 240V loads... right?  But wait.  What if  they are both on?  Say that water heater cycles on. 



This.  This is why the electrician and the power company read near-120V on both legs, declared victory and went home.  At the time they checked it, the water heater or other 240V load happened to be cycled on.   What they neglected to check was the voltage across the 2 hot legs, which should have read 240V but would have read perhaps 7 volts. 

I bet they actually read 122 and 115V, and told you the difference was normal. 

Next time they come out, switch all 240V and MWBC breakers off to stop that goofy flow pattern you see in the third diagram. Then, they'll be able to see the problem. 

But the dead loads aren't all on the same side of my panel!

Because you're presuming left and right are the two poles. That's only true on a Pushmatic panel.  Here's how most panels are actually laid out, and now see if that matches the location of your dead breakers.

This will fry your well pump!

If you look at the diagram carefully, you can see that the water heater is in series with the loads on the dead pole.  And, there's only 120V for both of them.  It all depends on how much load the water heater is drawing versus the other loads, but if the loading is modest, the voltage will be near 120V.  If you have 13A of 120V load, then it will be 60V to the water heater and 60V to the 120V loads.  

The water heater is perfectly fine with that, because it's a simple resistive load that will work on any voltage below spec.  It is able to slowly re-heat the water in the tank even on 1/4 voltage (1/16 wattage), and eventually it shuts off.  That's how well-insulated water heaters are.  

The well pump, on the other hand, will be one unhappy puppy.  Motors do not like this kind of massive undervoltage treatment, and it will burn out that motor PDQ.  Given where most people put their well motors (at the bottom of the well), you really don't want that.