Over the years I have purchased two 3.5 HP electric chain saws and although using them only infrequently for trimming trees and clearing storm damage, I have cooked them both. I now have ordered a new Poulan 3.5 HP electric saw and it should be here (home) 08-18-2009. It's rated 12 Amps.

I have cooked the motors because of undersized extension cords. These saws are not supplied with thermal overload protection nor a current fuse. I know that each 50-foot length of extension cord that I include decreases the voltage at the saw motor and increases the current. I also know that I should be using 10 AWG extension cords, but it's simply less expensive to purchase another saw.

I don't understand the 12 Amp rating because 12 x 110 / 745 = 1.8 HP (not 3.5 HP)
I could sure use your enlightenment with that.

Anyway, I'm thinkig about mounting a 10 Amp circuit breaker in a plastic watertight Bud Box. The box will include a male panel mount plug that my extension cord will plug into. On the other end of the box will be approximately 6-feet of power cord and a female receptacle that will plug onto the chain saw. As such, I am now limiting the current at the saw to 10 Amps.
Will this prevent me from cooking my new saw?

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Yes 3.5 HP is unrealistic.

What size cord(s) are you using and what length(s)?

Cooling is usually the problem. Sawdust may block the holes. Overloading is yet another reason.

A thermal overload in the motor would be best.

A 10 A fuse might pop even if it's time delay.

A motor overload might work.

Keep compressed air handy to clean out the cooling passages.

A small gas powered chain saw would eliminate all your problems... except those defined as "user error" I suspect that a dull chain will greatly overload an electric chain saw even in the simplest of tasks. If you try to cut a limb over 2 inches you are pushing the limits of an electric chain saw. For small limb triiming you would be better off with a reciprocal saw with an appropriate blade.

Unless you are using something like a 18 gauge, 100" extension cord, it is unlikely the cord is the problem. Even then you would have to use it constantly for an extended period of time.

I do not wish to be unkind, but more likely it is you cutting technique. If you have a dull chain and are really forcing it you could, over work the motor. Logically that should tire you out before the saw over heats.

With a sharp chain you should only apply slight downward pressure, letting the saw do the work.

I have a small, cheap, Remington, electric saw that I use on the end of a 14" pole. Needless to say that it has to be small, to be used on the pole. I think it has a 12" bar.
It will easily cut a 6 to 8" limb.

Most likely problem here is an improperly tensioned chain. Chain should be such that you can lift the chain about 1/2 off the bar in the center of the bar. An overly tight chain would put unnecessary, constant load on the saw motor. As the bar heats up and expands the tension on the chain increases.

The 3.5 HP is some kind of sales talk, probably "a developed horse power at a xxx rpm", what ever that means.

Your absoluely right about cutting technique and a sharp blade and the chain needs to stay oiled too.

You have to cut, so the saw won't bind. I think it's cut about 1/3 in on an angle on the bottom of a branch that needs to be cut.

The sideways cut prevents the tree from collapsing onto the saw. Then cut that section out straight. It's still unlikely that the branch will pinch the saw.

Now cut from the top and the branch will follow the wedge.

Don't skimp on the step and always cut pieces of the branch making it smaller and smaller. Only for small branches can you cut directly, but you won't know where they will fall.

Yep, I use an electric one too. Occasional trimming. It's about 30 years old. I have to replace the oil bladder and cord strain relief. Have the parts. Don't have the time.