I am aware of the 'one sigma', 'two sigma' etc 'level of significance'. However this description of the significance of a result with sigmas seems to say 'If this other theory was not correct, then there would be only an $x\%$ chance of this experimental result'. For example, with the Higgs boson 'discovery', the result was significant to the 5 sigma level because there was only a one in 3.5 million chance of the Higgs boson-like 'hump' that was seen simply being part of the background noise.
Suppose you are trying to test how closely a value predicted by a theory matches experiment. You perform the experiment and get, say, 0.998$\pm$0.014 and the predicted value is 1. Is there anything that you can say about this particular data to judge how closely the experiment fits with the data (e.g. can you divide the error up and say that the experimental value is within a quarter of a standard error)? Or are you simply limited with this given data to saying that the experiment is consistent with the theoretical prediction within the error, and then to judge more closely how well the prediction matches with the real world you would need to conduct higher precision experiments or take more repeats to reduce the standard error in your experimental value?