Melissa,
Your target heart rate zone depends on your Max Heart Rate. Each zone is a band - say 60% to 70% of Max HR.
The HRM may be accurate. It is the Max HR computation for you that could be wrong.
The default Max HR in most HRMs and equipment is based on the Age-Predicted formula. The Age Predicted Formula was not intended even by the creators of the formula to be accurate. It is a rough guideline of the average max heart rate of people based on age.
The Formula simply is Max HR = 220 - Age
The issue is that the Max HR of a person also depends on fitness level. If you exercise regularly your Max HR is likely to be higher than someone else your age. An athletes's Max HR is likely to be higher than the MAx HR of someone who exercises just enough for general fitness who is the same age.
Some HRMs allow you to customise Max HR. This may entail taking your resting HR and entering it into the HRM. The rationale is people who are more fit have a lower resting Heart Rate than people who dont exercise. If your HRM allows this, then it adjusts the Age Predicted MAx HR by bumping it up or down based on whether your resting heart rate is high or low. This is more accurate that the smple 220-Age but is still not accurate enough because resting rate rate is again a guideline.
There are more accurate Max HR tests that certain gyms will do for your based on a treadmill test. It will include pushing to your anerarobic threshhold and measuring fitness by how quickly your heart rate returns to normal after exertion.
In summary, the accuracy of the zones displayed by an HRM depends on whether it is using your true Max HR to calculate your zones.