That's a very curious problem, dataDrone. Apparently, the module will only succeed if it partly fails. For example, just switching the order of your headers makes the silly thing work.:

#!/usr/bin/perl -w
use strict;

my $row;

use HTML::TableExtract;
$/ = "";

my $string = <DATA>;
my $headers = ['Date of Incident','Date','Unit cost'];

my $te = HTML::TableExtract->new(headers => $headers) ;

$te->parse($string);

 # Examine all matching tables
foreach my $ts ($te->table_states) {
  print "Table (", join(',', $ts->coords), "):\n";
  foreach $row ($ts->rows) {
     print join(',', @$row), "\n";
  }
}

# use Data::Dumper ;
# print Dumper($te);

__DATA__
<TABLE>
<tr>
<th>Date</th>
<th>Date of Incident</th>
<th>Unit cost</th></tr>
<tr>
  <td>11-19-2001</td>
  <td>11-14-2001</td>
  <td>.01</td>
</tr>
<tr>
  <td>11-18-2001</td>
  <td>11-14-2001</td>
  <td>.01</td>
</tr>
</TABLE>
[download]

The explanation might be found if you uncomment those Data::Dumper lines above, and examine the output, where the pattern to be tested is printed out. A match is tested against this pattern:
'hpat' => '((Date of Incident)|(Date)|(Unit cost))'
But I gave it no further thought beyond this. Good luck.
mkmcconn