Yes, there is a chance.

And yes, the chance is indeed abysmally low, at least in my personal experience. I got one shoulder piece in about 40 or so silver keys that I used. I know people who have gotten much better drop rates from silver, though. The drop rates from gold is far better. About 1-in-3 in my personal experience.

I don't know how it is on consoles, but there are a lot of VR16s on the PCs who group up with low-level VRs to do the dungeon at a low scaling if they don't feel confident about the vet dungeons. I saw a lot of them back when my low-level VRs went looking for PUGs. Ideally, you would've started trying the vet dungeons at VR1 and by the time you hit VR16, you would've had enough practice and experience to master both the dungeons themselves as well as how you play your build. It's a good way to tune your build (I made several drastic changes to my build from VR1 to VR16, largely from the learning experience of the harder content). But, as they say, better late than never. Be open to the idea of changing and evolving your build as you spend time with the vet dungeons.

Also, I used to look for PUGs around the same consistent time each day, and after a couple of weeks, I got to know a lot of the other people who do pledges at around that time. These days, I'm always running with the same group--people that I originally met in PUGs each day--and we clear both pledges at VR16 in 40-50 minutes total each night.