The only time LinearFit iterates to attempt to do a fit is when the data has errors in both coordinates. In this case, if the fit has not converged in MaximumIterations, the fit stops and a warning message is generated. Experience has shown that very rarely will a fit require a default value of MaximumIterations equal to 15. Nonetheless, the limit may be increased if required. Here is what the relevant part of the Fit Options screen looks like to set this option: