Most estimators and filters provide assessments of their own estimation error, often in the form of mean-square error. Are these self- assessments trustable? What is the degree to which they are trustable? This is Part I of a two-part series that provides answers to some of these questions, referred to as the credibility of the estimators. It formulates the concept of credibility, proposes tests for MSE-credibility, and discusses their superiority to the existing test. Numerical examples are provided to illustrate the utility and effectiveness of the proposed tests and the drawbacks of the existing test