As long as your input files are not too large it might be easier to slurp the entire file into memory as a string and capture each document in turn using a global regex match. Here, I capture the document in $1 and, within it, the case name in $2. I have put the data (including a second document) after the __END__ token in the script so I can read it with the <DATA> filehande which Perl automatically opens for us.

use strict;
use warnings;

my $allDocs = do
   {
       local $/;
       <DATA>;
   };

my $rxExtractDoc = qr
   {(?xms)
       (
           \d+\sof\s\d+\sDOCUMENTS
           .*?
           No\.\s(\w+)
           \n
       )
   };

while ( $allDocs =~ m{$rxExtractDoc}g )
{
    my $document = $1;
    my $caseName = $2;
    my $edCount  = () = $document =~ m{EDUCATION}g;

    print
       qq{Case Name: $caseName\n},
       qq{Education: $edCount\n},
       q {-} x 25,
       qq{\n};
}

__END__
Blurfl Blurfl

No. 32323IY883

97 of 141 DOCUMENTS

HOKE COUNTY BOARD OF EDUCATION; HALIFAX COUNTY BOARD OF
EDUCATION; ROBESON COUNTY BOARD OF EDUCATION; CUMBERLAND
COUNTY BOARD OF EDUCA\TION; VANCE COUNTY BOARD OF EDUCATION;
RANDY L. HASTY, individually and as guardian ad litem of
Randell B. Hasty; STEVEN R. SUNKEL, individually and as
guardian ad litem of Andrew J. Sunkel; LIONEL WHIDBEE,
individually and as guardian ad litem of Jeremy L. Whidbee;
TYRONE T. WILLIAMS, individually and as guardian ad litem of
Trevelyn L. Williams; D.E. LOCKLEAR, JR., individually and
as guardian ad litem of Jason E. Locklear; ANGUS B. THOMPSON
II, individually and as guardian ad litem of Vandaliah J.
Thompson; MARY ELIZABETH LOWERY, individually and as
guardian ad litem of Lannie Rae Lowery; JENNIE G. PEARSON,
individually and as guardian ad litem of Sharese D. Pearson;
BENITA B. TIPTON, individually and as guardian ad litem of
Whitney B. Tipton; DANA HOLTON JENKINS, individually and as
guardian ad litem of Rachel M. Jenkins; LEON R. ROBINSON,
individually and as guardian ad litem of Justin A. Robinson,
Plaintiffs and ASHEVILLE CITY BOARD OF EDUCATION; BUNCOMBE
COUNTY BOARD OF EDUCATION; CHARLOTTE-MECKLENBURG BOARD OF
EDUCATION; DURHAM PUBLIC SCHOOLS BOARD OF EDUCATION; WAKE
COUNTY BOARD OF EDUCATION; WINSTON-SALEM/FORSYTH COUNTY
BOARD OF EDUCATION; CASSANDRA INGRAM, individually and as
guardian ad litem of Darris Ingram; CAROL PENLAND,
individually and as guardian ad litem of Jeremy Penland;
DARLENE HARRIS, individually and as guardian ad litem of
Shamek Harris; NETTIE THOMPSON, individually and as guardian
ad litem of Annette Renee Thompson; OPHELIA AIKEN,
individually and as guardian ad litem of Brandon Bell,
Plaintiff-Intervenors 

v. STATE OF NORTH CAROLINA and the
STATE BOARD OF EDUCATION, Defendants

No. 530PA02

98 of 141 DOCUMENTS

HOKE COUNTY BOARD OF EDUCATION; HALIFAX COUNTY BOARD OF
EDUCATION; CUMBERLAND COUNTY BOARD OF EDUCATION; VANCE
COUNTY BOARD OF EDUCATION; RANDY L. HASTY, individually
and as guardian ad litem of Randell B. Hasty; STEVEN R.
SUNKEL, individually and as guardian ad litem of Andrew J.
Sunkel; LIONEL WHIDBEE,
individually and as guardian ad litem of Jeremy L. Whidbee;
TYRONE T. WILLIAMS, individually and as guardian ad litem of
Trevelyn L. Williams; D.E. LOCKLEAR, JR., individually and
as guardian ad litem of Jason E. Locklear; ANGUS B. THOMPSON
II, individually and as guardian ad litem of Vandaliah J.
Thompson; MARY ELIZABETH LOWERY, individually and as
guardian ad litem of Lannie Rae Lowery; JENNIE G. PEARSON,
individually and as guardian ad litem of Sharese D. Pearson;
BENITA B. TIPTON, individually and as guardian ad litem of
Whitney B. Tipton; DANA HOLTON JENKINS, individually and as
guardian ad litem of Rachel M. Jenkins; LEON R. ROBINSON,
individually and as guardian ad litem of Justin A. Robinson,
Plaintiffs and ASHEVILLE CITY BOARD OF EDUCATION; BUNCOMBE
COUNTY BOARD OF EDUCATION; WINSTON-SALEM/FORSYTH COUNTY
BOARD OF EDUCATION; CASSANDRA INGRAM, individually and as
guardian ad litem of Darris Ingram; CAROL PENLAND,
individually and as guardian ad litem of Jeremy Penland;
DARLENE HARRIS, individually and as guardian ad litem of
Shamek Harris; NETTIE THOMPSON, individually and as guardian
ad litem of Annette Renee Thompson; OPHELIA AIKEN,
individually and as guardian ad litem of Brandon Bell,
Plaintiff-Intervenors 

v. STATE OF NORTH CAROLINA and the
STATE BOARD OF EDUCATION, Defendants

No. 534PA08

End of document set
[download]

Here's the output.

Case Name: 530PA02
Education: 11
-------------------------
Case Name: 534PA08
Education: 8
-------------------------
[download]

I hope this is of use.

Cheers,

JohnGG

---

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