His list of "symptoms" that a relational database is not right for you are more often symptoms that your relational database was not well designed (or that the problem you were trying to solve changed along the way) rather than that the relational model itself does not fit your needs.
I think this applies to all his "structural symptoms" but especially with "Do you have tables with lots of columns, only a few of which are actually used by any particular row?" This is more often a sign that the database was not properly normalized than anything else.
This of course is not to say that the relational model is right for everything. In some cases, object oriented databases are the way to go and if you truly need a vast level of scalability and you can afford to relax the ACID standard then it makes sense to look at non-relational options.
There is room in the world for both relational and non-relational models, but I do not think fashion should play a role in choosing a technology personally.