That seems like a pretty unwise assumption to me. In a decade or two the web may have changed beyond all recognition.
Programming may be completely different in a decade's time but I won't be programming. My era is now and that was always the plan to concentrate on web.
I fully respect the "career programmer" may need to choose differently but for many of us working as an engineer/programmer is simply a step in the journey. To go back to the OP, that's probably why I didn't feel inclined to study CS at university.
Lets say the next big thing in programming is GPU programming to compute massive volumes of data - I will still always do web programming.
Lets say I continue to move into strategy, product development etc - I will still always do web programming.
I just don't sit into one of the boxes perhaps you are trying to pigeon hole me into, maybe?
However, because I know I will always do web programming I have definitely not focused on issues like compiler design, low level memory management [1], etc because most of the languages I use are either interpreted or compiled to bytecode and thus those areas are irrelevant
I would therefore argue it very much depends on what kind of work you wish to do - sure read some more in depth stuff but if you are doing web programming spending some of that time on product development knowledge, server administration, etc might be more beneficial.
[1] clearly some memory management is important in languages like Java and Objective C but using the built-in alloc/dealloc functionality is not really "low level understanding"