No, the connector is straight on the PCB. It's all a really tight squeeze.
Edit: I think I got it working again.
The case's logic board had a sheet of plastic on it that I thought was meant to be removed because it made installing the drive an extremely tight squeeze. I removed it, and then the thing stopped working. So I jury rigged a sheet of plastic cut from a Ziploc bag, sandwiched it between the drive's PCB and the case's PCB, and now it's working ok.
The moral of the story is "don't remove protective plastic sheets." Also I think one of the computers I tested with has a funky FW port.