The trivial answer is yes, at least under certain circumstances, as the example you give shows.
First, the concept of substrings is not particularly useful in trademark law. IF that were the case, then we wouldn't be able to have a company called "Gaudiest Clothing Brand" because it contains the substring audi.
But, you might say, what about looking at words instead of characters, as with the example of "Linux" and "The Linux Foundation." In that case, I offer the example of "Target," which is a word found in literally thousands of US trademarks.
You'll notice that the first item on this list where word mark is simply target, rather than a phrase containing that word, doesn't even belong to the well known chain of stores. This brings us to another important aspect of trademark protection, which is that the mark's purpose is to identify a producer of goods, a provider of services, or particular goods or services themselves. You could probably start a bookstore called McDonald's, unless you're near Redmond, Washington, but you certainly would not be able to use that name for a chain of hamburger-based fast food restaurants.
Back to the subject of Linux, one possible explanation here is therefore that Linux is a trademark for an operating system, while The Linux Foundation is a trademark for a non-profit foundation that is concerned with the operating system. The trademarks don't represent competing entities. Consider, for example, the disclaimer on volkswagenownersclub.com:
VolkswagenOwnersClub.com is an independent media publication. VolkswagenOwnersClub.com and its owners are not affiliated with or endorsed by Volkswagen AG or Volkswagen of America, Inc. Volkswagen is a registered trademark of Volkswagen AG. All rights reserved. All information Copyright 2006-2010
(Also consider, for example, the case of Apple Corps, Ltd. and Apple Computer, Inc..)
As far as I can tell, however, "Volkswagen Owners' Club" is not directly relevant to this question, because it is not a registered trademark, and the question concerns two registered trademarks. I am not sure whether two such trademarks could coexist if there were an adversarial relationship between their owners.
In this case, hwoever, the trademarks are used together because the owners have a collegial relationship. They work for a common goal, the promotion of the product identified by one of the trademarks. The foundation's use of the trademark is therefore undoubtedly with permission. In fact, the Linux foundation is Linus Torvalds' employer, and one of the foundation's purposes is to manage the Linux trademark that Torvalds owns.
Now the question of derivative trademarks, such as My Linux is clouded by the fact that Linux is an open source operating system. In principle, you can't modify a product and then sell it using its original trademark unless you have permission. For example, I don't suppose I could rebottle Coca-Cola with some added salt and pepper and sell it as *Phoog's Coca-Cola" unless I had permission from the owner of the trademark.
With Linux being an open-source project, however, the terms of the open source license probably explicitly permit people to use the trademark under certain restrictions if they offer, for example, a customized distribution of Linux