Couple of things :)

(1) You need to localize you global filehandle FILE. Otherwise all filehandles stored away in the array will refer to the most recently opened file...

local *FILE;
open FILE, "<", "./$nbdc_dir/$file" or die ...
...
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Or, even better, make use of the feature of more modern versions of Perl to accept a lexical variable in the open() statement:

open my $fh, "<", "./$nbdc_dir/$file" or die ...
$nbdc_filehandles[$index] = $fh;
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or even simply (if @nbdc_filehandles is lexical)

open $nbdc_filehandles[$index], "<", "./$nbdc_dir/$file" or die ...
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The diamond operator is somewhat "special" syntactically, in that

(2) you need to make an explicit assignment if you use it outside of loops:

 my $line = <$fh>;

or

 $_ = <$fh>;

(3) no whitespace is allowed within the angular brackets:

$_ = <$fh>;    # OK
$_ = < $fh >;  # not OK
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(4) and, as jettero and BrowserUK pointed out, it doesn't accept array expressions, so you have to use an intermediate flat scalar, or readline():

my $fh = $nbdc_filehandles[$index];
$_ = <$fh>;
# instead of
# $_ = <$nbdc_filehandles[$index]>;
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So, your code would look like

my @nbdc_filehandles;
my @nbdc_data;
my $index=0;

foreach my $file (@nbdc_files) {
    open my $fh, "<", "./$nbdc_dir/$file" or die "Can't open ./$nbdc_d
+ir/$file";
    $nbdc_filehandles[$index] = $fh;
   
    #get the first set of data values
    
    $_ = <$fh>;
    $_ = <$fh>;
    chomp; #get rid of trailing \n
    $nbdc_data[$index] = [ split(/\t/, $_) ];
    # ...
    
    $index++;
}

# then sometime later, to reuse the stored filehandles

my $fh = $nbdc_filehandles[$idx];
my $line = <$fh>;
# ...
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