Good question.

I'll try to answer it with examples. Let say we have a DNA sequence:

ACGTACGTACGTACGTACGTACGTACGAGACTACGT

Now, for some unknown reason DNA break occurs:

ACGTACGTACGTACGTA | CGTACGTACGAGACTACGT

DNA ends can be joined back:

ACGTACGTACGTACGTACGTACGTACGAGACTACGT

In this example, no evidence of breakage is present.

ACGTACGTACGTACGTATACGTACGTACGTACGAGACTACGT
_________________^----^___________________

In this example, DNA breakage site has additional DNA inserted that is duplicate of one DNA end.

ACGTACGTACGTACGTATTTTTAAAAACCCCGGGCGTACGTACGAGACTACGT
_________________^---------------^___________________

In this example, DNA breakage site has additional DNA inserted and it is unclear were it came from.

This "unclear were it came from" DNA is called "non-template DNA".

DNA end joining mechanism determines such addition of extra DNA in break site (see, http://en.wikipedia.org/wiki/Non-homologous_end_joining).

In breakpoint sequencing there are problems where inserted DNA is longer than reads, because it's hard (maybe even impossible) to map breakpoint.

DNA:   ACGTACGTACGTACGTATACGTACGTACGTACGAGACTACGT
#                       ^^^^^^
Reads: ----------- ----------- ----------- ----------- ----------- -----------
             -----------   -----------  -------------   -----------  -----------

In this example reads mapped all breakage site (DNA ends + insert): CGTATACGTACGT - now we know that there changes in our DNA and we can still trace back were those changes come from.

DNA:          ACGTACGTACGTACGTATTTTTAAAAACCCCGGGCGTACGTACGAGACTACGT
#                              ^^^^^^^^^^^^^^^^^
Short reads:  ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- 
                ----  ----  ---- ----  ---- ----  ---- ----  ----

In this example, with such long insertion (that is non-template) it's hard to trace back native DNA site.

Edit

In order to solve this problem Sanger sequencing is used at the guessed breakpoint site.

See figure 4A from http://www.sciencedirect.com/science/article/pii/S0092867411008853 In breakpoint 6 (3105 strand): there is 64bp insertion of non-template (probably) DNA.