After trying the suggestions from other answers, consider that the numbers might be real. I do have the same problem and I returned my first HRM because I thought it was faulty. It turned out that I just do have unusually high heart rate without any known heart conditions, so that 150 is more or less normal for casual riding and the maximum is somewhere ...
Since no-one else mentioned it - this can be caused by the air causing some synthetic shirts/base layers to flap/flutter.
I had a top that got relegated to use on the indoor trainer because when wearing it outdoors my HRM would record 200+ any time I picked up any speed.
As mentioned already, one likely problem is poor contact.
One easy way to test for poor contact is to get the contact area wet - and I mean wet.
Put on your HRM strap and get your GPS unit. Turn on the GPS unit, and when it's reading your HR - or when it should be reading your HR - take a full water bottle and soak in between the HRM strap and your skin. ...
Try replacing the battery, it could be running flat.
Make sure you're getting proper contact with your HRM to your chest - try adding some conductive gel.
Try a different soft strap, they wear out after a while.
I've had inconsistent readings from all 3 of the above issues at times.
I haven't found any of these answers satisfactory. I use my road bike on my trainer, so fit can't be an issue. I don't train hard on the trainer, always 130bmp heart rate. The only difference is I spend more time upright on the trainer than on the road.