A bit of both. The market is white hot right now, so there's a better than usual chance of getting something. Of course there's still a lot of applicants so it's useful to have the right degree.

The thing to do is talk to headhunter firms. I've always maintained a number of relationships with the leading firms in my area (quant trading) and they'll call now and again. But if you don't know them, get in touch and they'll usually tell you if you're useful to them.

You can learn the basics of FPGA development on your own (I did), but you need to go to a company that uses them to actually get good.[0] Which probably means a defense contractor or maybe a telecom company. A finance firm isn't going to hire some random person who knows FPGA dev unless they have done real world projects.

[0]: I'm sure there are exceptions of people who self taught themselves and are great, but it is significantly harder to break into than software.