Are you sure that you don't have another library of the same name?
I search on my computer for "Fat16.h" and i found in two location:
...\My Documents\Arduino\libraries\Fat16-master\Fat16...\Program Files\Arduino\libraries\Robot_Control
I deleted the Robot_Control folder and after I try to compile the fat16write.ino and i get these errors:
fat16write:12: error: 'SdCard' does not name a type
fat16write:13: error: 'Fat16' does not name a type
fat16write.ino: In function 'void error_P(const char*)':
fat16write:19: error: 'PgmPrint' was not declared in this scope
fat16write:20: error: 'SerialPrintln_P' was not declared in this scope
fat16write:21: error: 'card' was not declared in this scope
fat16write.ino: In function 'void writeNumber(uint32_t)':
fat16write:40: error: 'file' was not declared in this scope
fat16write.ino: In function 'void setup()':
fat16write:46: error: 'PgmPrintln' was not declared in this scope
fat16write:50: error: 'card' was not declared in this scope
fat16write:53: error: 'Fat16' has not been declared
fat16write:53: error: 'card' was not declared in this scope
fat16write:63: error: 'file' was not declared in this scope
fat16write:63: error: 'O_CREAT' was not declared in this scope
fat16write:63: error: 'O_EXCL' was not declared in this scope
fat16write:63: error: 'O_WRITE' was not declared in this scope
fat16write:65: error: 'file' was not declared in this scope
fat16write:66: error: 'PgmPrint' was not declared in this scope
fat16write:71: error: 'file' was not declared in this scope
fat16write:78: error: 'file' was not declared in this scope