Somewhat side-tracked in writing these (pt1, pt2, pt2.5) and thinking about how best to explain some of the issues, especially when getting to the deeper semantic levels. However a work discussion about Apple and Amazon's security flaws and the case of Mat Honan provided an interesting answer which I think describes the problem quite well.
In the above incident hackers used information from Amazon to, primarily, social engineer Apple's customer service into believing that the hackers were Mat Honan. From the Wired article (linked above and ), Honan provides the quote:
But what happened to me exposes vital security flaws in several customer
service systems, most notably Apple’s and Amazon’s. Apple tech support
gave the hackers access to my iCloud account. Amazon tech support gave
them the ability to see a piece of information — a partial credit card
number — that Apple used to release information. In short, the very four
digits that Amazon considers unimportant enough to display in the clear
on the web are precisely the same ones that Apple considers secure
enough to perform identity verification. The disconnect exposes flaws in
data management policies endemic to the entire technology industry, and
points to a looming nightmare as we enter the era of cloud computing
and connected devices.
Both accounts at Apple and Amazon have a user identifier and passwords; they also have a set of other criteria to establish whether a given human is who they say they are. In Amazon's case they ask for email, address and the titles of some books you've bought from them - at least last time I called. In Apple's case it looks like they wanted some more personal information, in this case the "least significant digits" of a credit card number. I say least significant because these particular digits are often printed in plain text on receipts. As far as Visa, MasterCard, Diners etc consider, these digits have no meaning - though I have an issue with that as we shall see.
In Amazon's context the data about a user is semantically isolated from Apple's context. This is a level deeper than saying that they both had user identifiers but to what Real World concept and instance those identifiers meant and represented. The trouble here started when the realisation was made that the instance that the Amazon identifer related to could be the same as the thing that the Apple identifier related to, in this case the Real World Mat Honan. To make this complete it also turns out that the meaningless four least significant credit card digits in Amazon's context were the proof of identity in Apple's context.
We can argue that the data and identity management procedures in both cases were at fault, however in analysis this was actually hard to see: How could 4 random digits, effectively uniquely identify a person and without an understanding of each other's semantic view of the world, who would have realised?
The whole hack described in Mat Honan's article goes into a lot more detail on how this information was found out. Indeed much of the information required is already public and it is just a case of putting all this together and effectively using that to make a consistent profile.
As for credit card numbers, the practice of displaying the final four digits or the "least significant digits" in certain semantic contexts is called PAN truncation. However as the whole number has a well defined structure (ISO/IEC 7812), check summing and only a limited number of options for the rest of the digits is becomes feasible to reconstruct much of the number anyway, especially as some receipts also print the card type - at least enough to sound convincing in a social engineering situation if necessary. Furthermore as described in the article faking credit card numbers because of their structure actually now becomes a method of generating data to prove identity in some cases. In summary, there are no "random digits" or "least significant digits" in a data structure with particular meanings associated with each part of that structure.
The situation gets worse when more information can be provided for the social engineering exercise: for example, in Finland it used to be common before chip and pin terminals for a shop cashier to ask for identity, where the customer would show a "valid identitiy document" (this varied by cashier, shop and day-to-day in some cases) and certain details would be written down: usually the last four (least significant aparently) digits of a Finnish social security number or a whole passport number plus other varied details depending upon the shop and phase of moon etc.
 Mat Honan. 2012 How Apple and Amazon Security Flaws Led to My Epic Hacking. Wired Aug 6, 2012