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PBS interviewed John Rennie, the editor of Scientific American magazine, and got a somewhat better answer, if an obvious one.

In this case, unfortunately, what happened was that the surge of water associated with Hurricane Katrina moved through the lake, struck the levees and opened up these holes in a few places allowing the waters from the lake to then start to flow down into the city…

New Orleans has been hit by a lot of hurricanes over the years and the levees really are constructed so that they can withstand a lot of the sorts of pressures and strains associated with typical hurricanes.

The fact is that even if Katrina had really hit New Orleans dead on and if we’d seen the kind of 25-foot surge that was associated with other parts in the worst part of the storm, if that had hit New Orleans, even if the levees had held up, an enormous amount of water would have still spilled over them and flooded the city.

So really nothing is built to withstand something the strength of Katrina. But the fact is that in practice, New Orleans didn’t have to experience that extraordinary level of force. It was still enough though to break open these areas of the levees.

The interview pointed me to this article detailing how New Orleans got into this mess, and how a $14 billion marsh-restoration project might have started getting it out:

Restoring coastal Louisiana would protect the country’s seafood and shipping industries and its oil and natural-gas supply. It would also save America’s largest wetlands, a bold environmental stroke. And without action, the million people outside New Orleans would have to relocate. The other million inside the bowl would live at the bottom of a sinking crater, surrounded by ever higher walls, trapped in a terminally ill city dependent on nonstop pumping to keep it alive.

"Terminally ill" sounds about right.

This report indicates that the storm surge that hit New Orleans was larger than its classification (by wind speed) as a Category 4 storm would indicate:

Katrina weakened slightly to a Category 4 hurricane with maximum sustained winds estimated at 145 mph as it made landfall early Monday, but it maintained a storm surge that is only generally found in category 5 storms.

It’s the storm surge, not the wind, that is most relevant to the levee breaks.  So the levees, which were meant to withstand a Category 3 storm, really got a Category-5-type beating.

Some numbers begin to appear:

The lake, which normally is 1 foot above sea level, peaked at 8.6 feet above sea level, said English.

How tall are the levees?

Engineers speak:

"Levees fail. People need to realize when they make a decision to live
behind levees that there’s a risk that comes with that,” said Jason Fanselau
of the U.S. Army Corps of Engineers’ Sacramento office. "They can fail on warm
sunny days like we saw last year with Jones Tract (San Joaquin County), and
they can fail with huge wet storms like this week” along the Gulf of Mexico.

This part doesn’t sound right: 

The Category 4 Hurricane Katrina caused two levee collapses  —  a
300-foot breach in the 17th Street Canal flood wall and a smaller break in the
Industrial Canal flood wall  —  when water overtopped the floodwalls.

I don’t see how water overtopping the floodwalls necessarily causes the collapse—that would have been an entirely different problem.    Or is the top of the "floodwall" not synonymous with the top of the levee?

Here’s someone at Daily Kos who blames vortices that erode the levee from the bottom:

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Water always moves. When it comes up against an earthen barrier its
movement shifts from straight on or alongside the levee and is
transformed into a swirling motion.  Any variation in the levee — a
dip, a swerve, a slight indentation — catches the swirling currents
far below the top of the levee itself. The water swirls down there like
a giant very efficient drill, weakening the levee from within. When it
goes, it can do so quite suddenly; it "blows" a hole in itself and thus
the name "blew holes,"  sometimes called "scour holes." 

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Water only moves—the term he really wants is "flows" — when it’s not at mechanical equilibrium.  And the direction it’s flowing matters.  Did the water alongside each levee, in fact, have a net velocity that created—not just vortices—but vortices that were unanticipated by the designers  digging away at the base of the levee?

Erosion could be one cause.  What others are there?

A brief search for levee failure in the Science Citation Index turned up a number of references to levee failure.  Most of the ones I saw had to do with two mechanisms of failure (not counting overtopped levees—only broken ones.)

The first is percolation of water through porous layers of material in the levee, apparently referred to by some geotechnical engineers and others as "piping:"

A phenomenon called "piping" sometimes occurs under levees and may be accompanied by the formation of sand boils at locations where seepage surfaces.   Starting from the point of sand boil formation, a pipelike opening develops below the levee base … and proceeds toward the stream.  If this process continues, the failure of the levee becomes inevitable.   Permeability is a key component of piping models in defining the critical head limits needed for the safety of levees.  (C. S. P. Ojha et al., "Influence of Porosity on Piping Models of Levee Failure," J. Geotech. and Geoenvir. Engrg., Volume 127, Issue 12, pp. 1071-1074, December 2001, not available online to the general public.)

"Head limits" are pressure limits.  A direct proxy for pressure is the water level above the base of the levee.  A levee can fail to do its job "because the water got too high" for two reasons:  (1) the water spilled over the top of the levee, or (2) the water pressure difference across the base of the levee grew to exceed the threshold at which water can percolate through the levee material.

A combination of erosion and seepage seems to have been the main mechanism of structural failure during the Mississippi River flood of 1993:

Levee failures within the study reach occurred primarily as a result of overtopping linked with wave-induced surface erosion rather than structural failure, but a number of saturation-induced mass failures were precipitated by the high flood stages… In particular, side-slope seepage and underseepage were recurrent problems along the Sny Island levees, and were probably responsible for its eventual failure.  (Gomez et al., "Floodplain sedimentation and sensitivity," Earth Surface Processes and Landforms, 22 (923-936), 1997, not generally available online.)

Scouring is mentioned, too.

The other mechanism of failure that seems to be mentioned, but mostly by mathematicians, is hydraulic shock waves.  Is that even plausible?