Yes, the owner of the first issued patent would try to get the other 5,303 patents into a reexamination process at the USPTO if the owner thought his/her IP was valuable enough. This alone makes the assertion of such "overlap" questionable.
Sticking to facts - on the audio clip the expert said there were (by some undisclosed search method) 5,303 things going through the system at the same time that had an overlap.Breaking that down - that is not saying that all 5,303 applications made it out as issued patents or taking into consideration that those that were eventually granted may have been required to make claim amendments before being deemed worthy.
Then there is the word "overlap". What patents cover is laid out in their claims. The territory covered by a claim is notoriously difficult to determine by a human reader. My opinion is that whatever overlap meant it could not have meant "claims that cover the same territory." That would be way to hard for a computer to determine.
This is claim 1 of the patent in question:
An online computer system providing commercial backup services to
remote customer computers over the Internet by performing the
(a) providing at least one remote storage area
for use in storing customer backup information;
(b) establishing a
first online Internet session with a customer's computer;
allowing the customer to sign up for backup services over the first
online Internet session, including the step of establishing a customer
identifier and associated password for the customer;
establishing a second online Internet session with the customer's
(e) requesting the customer to input the customer
identifier and associated password established by step (c);
validating the customer identifier and password requested by step (e);
(g) conditioned at least in part on validating step (f), allowing
the customer to access the remote storage area over the second
Internet session substantially as if the remote storage area was a
backup storage device physically and/or locally attached to the
customer's computer, including the steps of:
(1) encrypting backup
data provided by the customer's computer,
(2) transmitting the
encrypted backup data to the online backup service provider over the
second online Internet session, and
(3) storing the backup data in
the remote storage area provided by the online backup service
It is a very long claim with many required steps. For someone to infringe this claim they need to do everything the claim says. It is not broad. When the voices on This American Life say it seems to cover so many things they are noticing that many things are described. That is not "covering". Only the claims state what the inventors see as new and non-obvious.
Following up on this topic I went to the site of the firm that was used to do the analysis that produced the 5,303 number. In their detailed report they do analyze the claim of the patent in question to the two other patents they boiled the 5,303 list down to.