Can you mountain bike on an e-bike?

Yes, on a purpose-built e-MTB with suspension, a mid-drive motor, strong brakes and trail tires, on trails where e-bikes are allowed. Commuter and cheap hub-motor e-bikes are not built for real trails.

Are fat tire e-bikes good for mountain biking?

Usually not for technical singletrack. Most budget fat-tire e-bikes have hub motors, basic suspension and brakes, and often a throttle that makes them illegal on mountain bike trails. They work well on sand, snow and dirt roads.

What class of e-bike is allowed on mountain bike trails?

In the US, where e-bikes are allowed on trails, it is usually Class 1 only: pedal assist up to 20 mph with no throttle. Many non-motorized trails on federal land do not allow any e-bikes.

Is a hub motor or mid-drive motor better for mountain biking?

Mid-drive. It keeps weight low and central and uses the bike’s gears, which makes steep climbs and handling much better.

Are e-bikes good for mountain biking? Which ones are, and which are not

The short answer

Yes, if it is an e-MTB: a mountain bike designed around a motor, with proper suspension, a mid-drive motor, strong brakes and trail tires. Those bikes climb technical terrain that most riders would walk and descend as well as many regular trail bikes. No, if it is a commuter, cruiser or cheap "fat tire" e-bike with a hub motor and basic parts. Those are fine on gravel paths and will be outclassed, and sometimes broken, on real trails.

The second question is just as important: whether you are allowed to ride an e-bike on the trails near you.

What makes an e-bike good on trails

A mid-drive motor. Mounted at the cranks, it keeps weight low and central and works with the gears, which matters on steep climbs. Hub motors in the wheel make the bike harder to handle off-road.
Real suspension. Most e-MTBs are full suspension with around 140 to 170 mm of travel. A heavy bike needs it more than a light one.
Strong brakes. Four-piston hydraulic brakes and large rotors are standard on good e-MTBs, because more weight takes more stopping.
Trail tires. Sturdy casings and grippy rubber. Thin casings puncture easily under e-bike weight.
Trail geometry. A slack head angle and long wheelbase for stability at speed.
Pedal assist only. In the US, trail-legal e-bikes are usually Class 1: assist only while pedalling, up to 20 mph, no throttle.

What e-MTBs are better at

Technical climbs. Steep, rooty or rocky climbs become rideable, and you can practise them instead of pushing.
More descending. Several laps in the time one would take.
Mixed groups. Riders of very different fitness can ride together.
Stability. The weight keeps the bike planted on rough descents.

What they are worse at

Playfulness. Heavier bikes are harder to jump, manual and pop off small features.
Carrying. Hike-a-bike sections, stairs and car racks are all harder.
Running out. With a flat battery, you are pedalling a very heavy bike home. Plan rides with range to spare.
Cost and wear. More expensive to buy and quicker to wear out drivetrains, brakes and tires.

A fuller comparison is in e-MTB vs regular mountain bike.

Where you can ride one

This is the part that catches people out. In the US, access depends on who manages the land:

Many non-motorized trails on federal land treat e-bikes as motorized vehicles and do not allow them, unless the land manager says otherwise.
Some state parks, local trail systems and bike parks allow Class 1 e-MTBs. Others do not.
Motorized trails generally allow e-bikes.

Always check the specific trail system before riding. Riding where e-bikes are banned is the fastest way to lose access for everyone.

What about cheap fat-tire e-bikes?

Fat tires look rugged, but tires alone do not make a mountain bike. Most budget fat-tire e-bikes have a hub motor, limited or no rear suspension, basic brakes and frames not designed for drops and jumps. They are good for sand, snow, gravel and dirt roads. On technical singletrack they are slow, heavy and easy to damage, and many have throttles that make them illegal on mountain bike trails anyway.

Bottom line

A real e-MTB is a very good mountain bike, especially for climbing and for riding more. Buy one built for trails, check where you are allowed to ride it, and ride it with good manners. Then enjoy explaining it at the trailhead, or let the Watts tee do it.