Two data sets may not line up when transforming from NAD 1927 to NAD 1983 in the United States. This is not a problem when converting from NAD 1983 to NAD 1927.
The NADCON grid is automatically loaded into memory when ArcMap is started. If the grid is not loaded into memory, the results differ by less than a meter.
Make certain NAD 1927 is listed in the 'Convert from' box and NAD 1983 is displayed in the Into dropdown list.