Quick'n dirty. Uses the fact, that Test::More::builder() is not overridden.

use strict;
use warnings;

use Test::More;


#-- builder() does not exist in Test::More, just intercept call hierar
+chy...

sub Test::More::builder {
  my $limit  = $Test::More::BAILOUT_LIMIT;
  my $tb     = Test::Builder::Module::builder(@_);
  my @tests  = $tb->summary;
  my $passed = grep { $_ } @tests;
  my $failed = @tests - $passed;

  if ($failed >= $limit) {
    $tb->BAIL_OUT("Limit of $limit errors reached (passed=$passed, fai
+led=$failed)!");
  }

  return $tb;
}

#-- here, we set the limit
$Test::More::BAILOUT_LIMIT = 2;  # or i.e. ... = $ENV{BAILOUT_LIMIT}

ok( 1,    "okay");
is( 1, 2, "oops");
ok( 0,    "test $_/3 will fail..." ) for (1..3);
done_testing;
[download]

Result:

ok 1 - okay
not ok 2 - oops
#   Failed test 'oops'
#   at ./tm.pl line 29.
#          got: '1'
#     expected: '2'
not ok 3 - test 1/3 will fail...
#   Failed test 'test 1/3 will fail...'
#   at ./tm.pl line 30.
Bail out!  Limit of 2 errors reached (passed=1, failed=2)!
[download]

Can be done properly by deriving your own class, but you get the idea. You need one more test than the limit, though. The interception happens before the next test case is executed. Limit = N demands N+1 tests, but I guess that is acceptable...

Edit: This implements an individual per-test-file-limit. If you want the limit to encompass the whole test session, you'll need to maintain the counter's persistency and life-cycle across all sessions (i.e. using Storable).

++ This is awesome.

Although I didn't put in much effort at all, I have considered and very lightly attempted to achieve what OP is asking a few times over my 20 years coding Perl.

I'm intrigued by the "most of the time" parenthesis. Could you elaborate?
map{substr$_->[0],$_->[1]||0,1}[\*||{},3],[[]],[ref qr-1,-,-1],[{}],[sub{}^*ARGV,3]

Most of the time means that any tool written with Test2 in mind, or directly on Test2 will work as expected, this should include most methods on Test::Builder itself. However cpan tools written against old Test::Builder that do not use Test2's 'Context' system may still call $tb->ok and $tb->diag separately, in which case the diag will still be lost.

In Test2 you would do this:

    sub mytool {
      my $ctx = context();
      $ctx->ok(...);
      $ctx->diag(...);
      $ctx->release; # This is where the bail-on-fail is triggered
    }
[download]

The TB version:

    sub mytool {
      $tb->ok(...); # BailOnFail is triggered here
      $tb->diag(...); # Lost
    }
[download]

$tb methods do acquire context's, but each of those methods has to acquire it's own. That said the diagnostics TB itself generates in a failed-ok (like line number) will make the output, it is just extra diags are lost.

That is unfortunately a fundamental, and not-fixable limitation of Test::Builder that Test2 avoided with the context system. Test2 was specifically designed to overcome this and many other TB limitations.