The main potential issues with the engine are the IMS bearing, RMS leaking and bore scoring.
IMS (intermediate shaft) bearing - catastrophic failure leads to valves hitting the pistons and lunching the engine. I'm not sure if and when they updated this on the Cayman but on the 911 this was late 2005 which has dramatically reduced the incidence of failures. Unpredictable as to whether this will happen, but by no means a certainty! Well reported online due to the catastrophic nature of the failure - bearings can be replaced to stronger ones prophylactically.
RMS (rear main seal) - leaking oil seal between gearbox and engine. Keep an eye on oil levels and replace the seal of oil useage is ridiculous or a large puddle appears. By no means catastrophic if a small leak occurs but I wouldn't buy a car with a leaking seal unless it was priced accordingly. Replacing the RMS gives an opportunity to put in the stronger IMS bearing if it's not an updated Porsche one.
Bore scoring - oil starvation especially to cylinder 6. Much talk online, many horror stories. The worse offenders are the 3.8 911s with automatic gearboxes potentially because they set off in 2nd gear. Unpredictable but potentially preventable if the car is warmed up properly before thrashing it and regular oil changes are performed. I think the 3.4 is much less likely to get scoring due to different liners, but you can get the car inspected beforehand to put your mind at ease.
Youre looking at a 11 year old sports car at the end of the day. Expect to need suspension components and other wear items replacing. Buy carefully and these might have already been done, but you will need to maintain it as with any other car of this age.
Find an independent specialist who can perform a pre purchase inspection including looking at the cylinder bores. The £300 spent on this will be well worth it if it gives you peace of mind or averts an impending total engine rebuild!
At £16,000 you're able to buy one of the best handling and performing sports cars ever made. Buy it with your eyes open, look after it and drive the wheels off it. If you manage to find a good one for your cash you shouldn't loose too much money on it either.