In a standard electric HWH, the other thing that can go wrong is a failed thermostat. I've had that happen. There are two, upper and lower, one for each element, and only one element is powered at a time. From a totally cold tank, the upper tstat turns on its element to heat water in the upper portion of the tank, for quick availability of a limited amount of hot water. When that tstat is satisfied, it turns off its element by switching power to the lower tstat. The lower tstat will provide power to its element until satisfied, at which time it turns off its element's power.
After enough hot water is drawn, the lower element gets power first. With continued use of hot water (eg. teenager or wife using it), eventually the upper tstat senses loss of hot water in the upper portion of the tank and switches power from the lower tstat to the upper element.
So, if you have some hot water, but it runs out fast, then likely the lower element or tstat has failed. If there is no hot water at all, even briefly, then either the upper tstat or its element has failed. As Slick said, you can disconnect power to an element and do a continuity test to see if it has failed. But you also can test a disconnected tstat for continuity from hot power side in to element side out.
If an element tests OK but its tstat has failed, the fix is even easier, as no draining of the tank is required and about the only tool needed is a screwdriver. When buying a new tstat, make sure to get the right one (upper vs lower), as they are different. BE SURE TO DISCONNECT POWER TO THE HEATER BEFORE FIDDLING IN THERE, OR YOUR HEIRS MAY GET THE PROPERTY SOONER THAN EXPECTED!!
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