Its just what I would expect to happen, largely irrelevant of what the specific issue is.
If it doesnt though it might be fuel pump for example, as it will be working harder in fifth than in 2nd to get to those rpm in the first place, so its useful background info to know.
Is the ECU light on to say it knows there is a fault?
Ignition is where Im going to throw my guess, so either plug leads loose on the coilpack, plugs themselves knackered or leaking from the rocker cover into the plug holes (or water getting down the same way)