Fun exercise, here was my approach, I make the assumption that the last block of data is your city_state_zip jumble and the first block is your customer, everything else in the middle is addressing info. Then I pull everything apart from its ends city_state zip then I have to make the horrible assumption that there is a state code... in either case, I can still put city_state together which you can see in the dump if you uncomment it. About as fancy as I care to get in 25 mins. Cheers.

use strict;
use Data::Dumper;
my @customers;

while(<DATA>){
  my %customer;
  my (@data) = split /[\t]/, $_;
  my $city_state_zip;
  my $which = 0;
  for( 1..$#data){
    $which = $#data -$_;
    $city_state_zip = $data[-$_];
    last if ( $city_state_zip =~/\w+/ig );
  }

  $customer{name} = $data[0];
  for( 1..5){
    if( $_ <= $which ){
      $customer{qq/address$_/} = $data[$_];
    }
    else
    {
      $customer{qq/address$_/} = "NULL";
    }
  }
  $customer{city_state_zip} = $city_state_zip;
  push @customers, \%customer;
}

foreach my $c ( @customers ){
    my ($city_state, $zip ) = ($1, $2, $3) 
            if $c->{city_state_zip} =~/(.*)\s+([\d\-]+)/g;
    my ($city, $state) = ($`, $1) 
            if $city_state =~/\s(\w+)$/;
    $c->{city} = $city;
    $c->{state} = $state;
    $c->{zip} = $zip;
    print "
    $c->{name}$c->{address1}  
    $c->{address2}   
    $c->{address3}   
    $city, $state  $zip\n";
}

#print Dumper(\$customers[0]);

__DATA__
BIGGLE EQUITY MGMT ASSOC.     PIGGLE BIGGLING PLAN     FUTURES I & II 
+    TWO LONDDDD CENTER     TRENTON NJ 97302-5115 
AGRICULTURAL LIFE INSURANCE CO     MAIN     111 ELEVEN AVE PO BOX 123 
+    MORRISTOWN NJ 97631-0633     
MILLENA BANC     MILLINER BANK & TRUST CO     456 THAUMATURGE AVENUE  
+   NEWSOME MA 07145-6316     
BRICKABRACK ADVISORS     LIBERTARIAN BANK     888 CRAZED CREEK ROAD ST
+E 11     MEMPHIS KY 77882-5394     
FOO COMMUNITY BANCORP     THE FOOVILLE SAVINGS BANK     18 FOOL AVENUE
+     FOO NY 10567-1735     
ELROND ASSET MANAGEMENT     BRA3255 CSHYTFD     ATTN SETTLEMENTS DEPT.
+     466 LURCHING AVENUE     NEW YORK NY 10347 
CRIPPLE CREEK INVESTMENT ADVISORS     GLIB TALKER     666 9TH AVENUE 8
+TH FLOOR     NEW YORK NY 10235-2302     
SEPARATIST STATE BANK     ATTN: JOSE KWAN     XYZ FINANCIAL CENTER    
+ 145 TALKING FAST ROAD     NEWBURGH NY 07083-2340
[download]

I wanted to add that, I think the way Perl allows you to develop solutions to real world problems like this so quickly and elegantly speaks volumes of praise for the mountain of programmers that built this language. I don't think I have seen a closer knit community of folks so willing to help other programmers learn and develop than right here on Perl Monks either. :0) Super!
JamesNC