If you mean: "Is there a generally accepted threshold for deciding whether a person's anxiety level has changed significantly," I think the short answer is "no."

One work-around might be to use the standard error of measurement as the basis for deciding whether the change is probably "real." I don't have the STAI Manual in front of me, so the numbers here will be hypothetical, but the reasoning won't change when the correct figures are substituted.

Let's say the internal consistency (split-half or alpha) of the STAI is .90 and the standard deviation of STAI scores in a reference group (standardization sample or your "in house" norms) is 20. Then the standard error of measurement will be 20 times the square root of the quantity, 1 - .90. That works out to about 6.3 points. Now, you make a decision about how stringent you want to be. If you want to be 95% sure that a given difference between two STAI scores is "real," then they need to be at least two standard errors apart - in this example, at least 13.6 points (call it 14).

If the question is whether an individual's answers can change over time, I think they can and do change. Mostly it is state anxiety that changes, in response to external stressors. Trait anxiety is supposed to remain stable, determined by qualities of the individual that do not change much.

In practice even trait anxiety may be seen to change. In my longitudinal study of 50 practitioners of Natural Stress Relief (NSR) meditation, for example, I saw a decrease in the trait anxiety of almost every single subject, since NSR effectively eliminates internal stress of all kinds (see www.nsrusa.org/about-stress.php for our definition of stress). It was telling that when I contacted the only subject who reported an increase of anxiety, he reported that his father had just died around the time he learned NSR.

The other two responses here correctly address the other interpretation of this ambiguous question: whether the STAI can be used to judge the anxiety level of an individual. The answer is yes, but you have to pay attention to what you want to measure: a clinical finding or a statistical one. The DSM, for example, is frequently used for clinical evaluation, yet is meant for use in coding statistical studies.