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Data using the NAD27 datum does not overlay data using the NAD83 datum. The x and/or y coordinates are off by hundreds of thousands or millions of map units.
When the National Geodetic Survey (NGS) established DATUM NAD83, it applied a false origin to the projection parameters so that NAD83 data would not overlay NAD27 data. This was intentional on the part of the NGS.
Change all data desired to overlay to the same datum. For more information see the book "Map Projections - A Working Manual" by John P. Snyder U.S.G.S Professional Paper 1395.