Started to assemble the first circuit board and run into a problem straight way. It turns out that there is a short circuit between the 5V and GND planes on the board. It’s taken a bit of time to track down where the problem was but managed to locate the issue at the cost of sacrificing a couple of the circuit boards.
The problem was that several vias don’t seem to have any clearance between them and the power planes. Looking at the files in a gerber viewer then it looks like the vias have clearance on the two inner planes as expected.
By shining a light under the board it’s possible to see where the vias have no clearance. The ones highlighted in green are correct and have no issue, the ones in red seem to be shorting out the inner two layers. I have confirmed this by drilling out these vias and confirming that the direct short between them has gone.