Here is a small change to concat the next line to the current if the current has <B in it (you may even want to change this to $_ .= <> if /</; just to be sure). Should solve your prob and avoids slurping in the entire file at once. You are also dropping the text on the bluebird line preceding the <bluebird> tag, not sure if this is what you want.

updated

updated so it actualy reads its data from its data block!

Regards,
R.

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

LINE: while ( <DATA> ) {
  $_ .= <DATA> if /<B/;
  next LINE if (m/\.{4}<B.*ird>/s .. /(\+\=\+)$/);
  print;
}
__END__

(Example 1)
This is Example 1
This is some text....<Bluebird>..
MBAAAEgAAAQAB
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+

(Example 2)
but sometimes it looks like this:
This is some text....<B
luebird>..MBAAAEgAAAQAoBA
AAQKAREDSCETRTBDFS
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+
[download]

update

I fixed this for a case where a line ends in something begining in <B other than a part of <Bluebird> and added another test case in the data. Also note the pattern match given by the OP matches <Blackbird> as well. Anyone know a more ellegant regexp to do /<(B|Bl|Blu|Blue|Blueb|Bluebi|Bluebird)$/ ?

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

LINE: while ( <DATA> ) {
  $_ .= <DATA> while /<(B|Bl|Blu|Blue|Blueb|Bluebi|Bluebird)$/;
  next LINE if (m/\.{4}<B.*ird>/s .. /(\+\=\+)$/);
  print;
}
__END__

(Example 1)
This is Example 1
This is some text....<Bluebird>..
MBAAAEgAAAQAB
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+

(Example 2)
but sometimes it looks like this:
This is some text....<B
luebird>..MBAAAEgAAAQAoBA
AAQKAREDSCETRTBDFS
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+

(Example 3)
This is designed to break it ha ha
this will make trouble<B
lackCrow> This is some text....<B
luebird>..MBAAAEgAAAQAoBA
AAQKAREDSCETRTBDFS
blaah,blaah
=+=+=+=+=+=+=+=+=+=+=+
[download]