We own a Z1U, which is basically the same body as far as I have heard. We've had it on many shoots now and so far it's rock solid.
I train my team to be pretty darn careful, but still, with 3 producers using it on a regular basis, we've had zero problems.
May it ever be so!

We've got 4 x Z1s, have to say the most reliable cameras we've ever had and they get a serious amount of use. We did get one destroyed, I can verify that they will not survive leaving them on your mates car roof when he drives off.
Bear in mind that all prosummer cameras are built down to a price. ENG cameras aren't but they cost way more but even then you'll see most of them inside Portabrace covers and those covers cost as much as a cheap consummer camera.
Still I cannot think of anything on the FX1s that we used to have that felt overly cheapish, no different to the build quality on the venerable old PD150s. Although the protective lens 'shutter' isn't that solid, the lens controls feel very smooth. The mounts for the shotgun mic on the Z1 do leave a lot to be desired, more of an afterthought I suspect but at least Sony finally put the XLRs where they belong, thank you Panasonic for showing them how it should be done with the DVX100!
Bob.

Still I cannot think of anything on the FX1s that we used to have that felt overly cheapish I can't either. These are well-built cams, especially for the $$ that they cost. We've got 3 Z1's, a couple A1U's, and 2 of the Z1's have been on the front of motocross bikes. Now that spot is reserved for the A1 without microphone.
Bob, I've removed the shotgun mount entirely, it fits in a case better. And I don't use on-camera mics anyway.

Did you get a genuine FX1? Mine is rock solid and heavy. It's all relative, I guess.

What have you been using that was built better? Was it in the same price range?

Hard to know what aspects of the camera build are worrying you. Transiting from using the very solidly constructed 16mm Arriflex cameras I didn't start back in horror because the FX (and Z1) has plastic in the housing. Pretty solid in my experience. If you want to see what they look like inside go to http://www.eidomedia.com/hdv/gallery.htm.

To my knowledge the FX1 has the same body material as my VX2000. Looks like plastic, but it's not. It's magnesium.

OK,,,,Im back on the prozac,,,, guess I was expecting a 3K cam to be built like a 100 K cam ,, yeah, pretty stupid.
Nice day here and going out to play with these puppies.

When I got my FX1 my first reaction was horror because it felt too light. instinctively I feel if something is high quality it should be heavy. When making a buying choice, all other things being equal, I say go with the weight. But this is a old fashined protocol and I guess the new light weight high strength moder materials have changed everything. It's proved a good solid camera for me.

Not have been using the Z1 that much but have no problems at all with it. There is an "empathetic feel" to it's performance that says it will work just fine as long as you don't beat on it with a hammer. It also seems to perform better in difficult lighting situations and I use it in full auto all the time.

JJK