For the data fitting problem under interval uncertainty, we introduce the concept of strong compatibility between data and parameters. It is shown that the new strengthened formulation of the problem reduces to computing and estimating the so-called tolerable solution set for interval systems of equations constructed from the data being processed. We propose a computational technology for constructing a "best-fit" linear function from interval data, taking into account the strong compatibility requirement. The properties of the new data fitting approach are much better than those of its predecessors: strong compatibility estimates have polynomial computational complexity, the variance of the strong compatibility estimates is almost always finite, and these estimates are rubust. An example considered in the concluding part of the paper illustrates some of these features.