Then the points layer does not have a defined coordinate system, or it has been defined incorrectly when using the Define Projection tool Define Projection—Help | ArcGIS for Desktop
If the XY coordinates in the spreadsheet were in the range of plausible values for longitude (-180 to 180 degrees) and latitude ( -90 to 90 degrees), then it should have been given a geographic coordinate system with an appropriate datum. The most widely use being the GCS WGS84, which is the default for coordinates derived from GPS data.
Often, people want the coordinates to be in a projected coordinate system (eg. a UTM projection) so they use the Define Projection tool to 'define' it as such. That doesn't 'make it so' to quote Jean Luc Picard. It just fouls things up. It must be 'defined' correctly, BEFORE it can be changed into a new coordinate system using the Project tool Project—Help | ArcGIS for Desktop
Steps in workflow are important... articulate exactly what you did to arrive at your situation should the above not provide a solution