The phenomenon you describe is a type of cognitive bias. This particular bias of stretching facts to fit your theory is called confirmation bias, defined as
The tendency to search for or interpret information in a way that confirms one's preconceptions. In addition, individuals may discredit information that does not support their views. (here)
the tendency to search for, interpret, favor, and recall information in a way that confirms one's preexisting beliefs or hypotheses, while giving disproportionately less consideration to alternative possibilities. (here)
These definitions don't explicitly mention making up facts to fit one's theory, but any person who did this could certainly be said to be biased toward confirmation. This suggest that the term can also be used to describe the (possibly accidental) fabrication of "confirming facts".