This could be a number of things.
However having dealt with marketing software before it is often not so easy as it sounds to tear down a subscription as it is to make one.
Quite often as the sign-up process involves you interacting with the web service you will be talking direct to the mail relays of the company in question - thus things can be done real time. Once you have been identified your details will be added to the database and used in future mailshots.
However when the company does a mailshot, typically they will not be sending the mail direct from their company mail servers, they will be using a bulk mailing company. This involves uploading the message once and modifications to the address database to be mailed.
The reason for using a bulk mail company are many, not least because most businesses dont have the money for / or have the infrastructure (software, internet connection, expertise and dedicated hardware) to bulk mail.
Another factor in choosing the bulk mailing companies is that the larger well established companies adhere to a code of conduct which is published and respected by the better commerical spam blockers - thus there is little chance of your subscriptions being blocked as the source is identified and trusted.
Pretty much all the ten day warning is for is because they wont update the database until the next mailshot (usually one a week or so) and whilst if they are not doing any mailshot you should not receive anything, it is possible that the mailing company has identified a mail as mis-sent and resends it.
I dont think there is as much sinister in the 10 days, its just mainly to stop people complaining on the odd occasion where the mail house has resent before their database is updated.
Cheers