Mantras almost always oversimplify their subject matter. This is a case in point.
The advantages of normalizing are more that merely theoretic or aesthetic. For every departure from a normal form for 2NF and beyond, there is an update anomaly that occurs when you don't follow the normal form and that goes away when you do follow the normal form. Departure from 1NF is a whole different can of worms, and I'm not going to deal with it here.
These update anomalies generally fall into inserting new data, updating existing data, and deleting rows. You can generally work your way around these anomalies by clever, tricky programming. The question then is was the benefit of using clever, tricky programming worth the cost. Sometimes the cost is bugs. Sometimes the cost is loss of adaptability. Sometimes the cost is actually, believe it or not, bad performance.
If you learn the various normal forms, you should consider your learning incomplete until you understand the accompanying update anomaly.
The problem with "denormalize" as a guideline is that it doesn't tell you what to do. There are myriad ways to denormalize a database. Most of them are unfortunate, and that's putting it charitably. One of the dumbest ways is to simply denormalize one step at a time, every time you want to speed up some particular query. You end up with a crazy mish mosh that cannot be understood without knowing the history of the application.
A lot of denormalizing steps that "seemed like a good idea at the time" turn out later to be very bad moves.
Here's a better alternative, when you decide not to fully normalize: adopt some design discipline that yields certain benefits, even when that design discipline departs from full normalization. As an example, there is star schema design, widely used in data warehousing and data marts. This is a far more coherent and disciplined approach than merely denormalizing by whimsy. There are specific benefits you'll get out of a star schema design, and you can contrast them with the update anomalies you will suffer because star schema design contradicts normalized design.
In general, many people who design star schemas are building a secondary database, one that does not interact with the OLTP application programs. One of the hardest problems in keeping such a database current is the so called ETL (Extract, Transform, and Load) processing. The good news is that all this processing can be collected in a handful of programs, and the application programmers who deal with the normalized OLTP database don't have to learn this stuff. There are tools out there to help with ETL, and copying data from a normalized OLTP database to a star schema data mart or warehouse is a well understood case.
Once you have built a star schema, and if you have chosen your dimensions well, named your columns wisely, and especially chosen your granularity well, using this star schema with OLAP tools like Cognos or Business Objects turns out to be almost as easy as playing a video game. This permits your data analysts to focus on analysing the data instead of learning how the container of the data works.
There are other designs besides star schema that depart from normalization, but star schema is worth a special mention.