electrobraid horse fencing?

The posts are about 16' apart. And are 2-3" wide.

Are you using treated wood posts? They wil work fine for the electrobraid, but are a little far apart for mesh. Unless you buy a fairly heavy gauge mesh, like that shown in the photo, it will sag if the posts are too far apart. I have similar mesh around my back yard with 4-5" posts 12' apart, and it still sags a bit.

The mesh is quite expensive and challenging to work with in long stretches, as it is hard to keep tight. The ground also needs to be level for it to sit well. You will likely find it costs less and serves just as well to put in more substantial posts and multiple strands of rope.

Yeah I've thought about that. I think I'll go with the electrobraid, maybe.
I think our posts are pressure treated-I'll have to ask my dad.

Even pressure treated posts at 2-3" aren't going to be terribly strong. You will also need to brace your corners or put in heavier and longer corner posts.

What kind of ground are they going into? Is it soft, rocky, clay? That will make a big difference too. My paddock used to be a hay field where the soil is rich and black with few rocks or clay. I've had at least one post come loose from the ground and so I had to take it out and cement it back in.

The picture you posted is basically what most of my property is. We just purchased it, and the previous owners had sheep. I added a hot line to the top, up high, and ended up having to add one right at the goats chest height, because he figured out he could climb over it. My livestock fencing is only 4ft tall (and not tensioned very well) The wood posts they used are about 4 inches in diameter.

The ground at our place is dark dirt and clay deeper down. Some of the posts can be 'wiggled' a little bit. Would that be a problem? How thick should the posts be?

You don't really want any wiggle in your posts. If they move, they need to go in deeper or you need to cement or tamp dirt around them to make them more secure. As to how thick, I prefer the 4-5" posts because I could safely tie any or my horses to these and they wouldn't likely pull it out or break the post if they pulled back. How heavy your posts need to be depends on what you are using for fencing and how far apart they are. If you are stringing 3-4 strands of rope, you can get away with a thinner post, but will need more of them. If the clay in your ground is shallow, you may have difficulty pounding in thinner posts without breaking them. In this case, you can use heavier posts, metal posts or auger your post holes.

As another alternative, you could put in heavier posts further apart (say 50') and then fill the gaps with lighter or step in posts every 12-15'. This isn't my preference, but doable.

I use step in plastic posts to cross fence inside my paddocks, but would never use these for the perimeter or for permanent fencing. Two strands of rope is a out all they can handle without bending.