An open with mode '>>' should position to end of file. You need to seek to the beginning in order to read from the file. I added two lines to your program and it now appears to work as you would like it to.

#!/usr/local/it/bin/perl -w

#
# Create sample file with two 
# instances of "FAILED" string
#
use Fcntl qw(SEEK_SET);

$RFILE="example_file";                                      
open(F1,">$RFILE") or die "cannot open $RFILE for creation!\n";
print F1 "FAILED\nPASSED\nPASSED\nFAILED\nPASSED\n";              
close(F1);

#
# Open file for Append AND read
#
open(RESULT,"+>>$RFILE") or die "cannot open $RFILE\n";
seek RESULT,0,SEEK_SET;

#
# Set an array to the filehandle and count the
#  number of lines with the "FAILED" string
#

my @result_array = <RESULT>;
$number_failures = grep /FAILED/, @result_array;

#
# Overall status is FAILED if any "FAILED" strings
# are found
#

$OVERALL_STATUS = ($number_failures > 0) ? "FAILED":"PASSED";
print "+-------------------+\n";
print "|                   |\n";
print "| EO_FAIL_03 $OVERALL_STATUS |\n";
print "|                   |\n";
print "+-------------------+\n";
close(RESULT);

<code>---
print map { my ($m)=1<<hex($_)&11?' ':'';
$m.=substr('AHJPacehklnorstu',hex($_),1) }
split //,'2fde0abe76c36c914586c';
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