Thanks for the response but I still don't quite understand what is happening.

Here is what I do. (1) I import a matrix A using A=Import"data.dat" If I type A[[1,2]] I get 0.992745. (3) If Itype N[A[[1,2]],16]] I get exactly the same answer as in (2), which in my opinion is strange since the result should be given to 16 digit precision. However, if I use the FullForm command, I get all the digits 0.99274468827 that were stored in the file data.dat.

Again, it is just SHOWING 6 digits for a machine number. Under the Preferences => Appearance => Formatting => Numbers you can set the number of digits, displayed by default.