Everyone tries to simplify the New Orleans situation.
1. It was a city (American sense of the word) beginning back in the 1820s, when French and Spanish and American entrepreneurs expanded the commercial side of the economy. It was built inland, on natural levees, because the French wanted their government safe from hurricanes.
2. Historically, the Mississippi River system flooded every year, water went over the banks into low-lying prairie, woodlands and fields all along the rivers.
3. In the last third of the 19th c., attempts began to control the flooding upstream. Levees and dikes were built along the Miss. R. and its tributaries, funneling more and more of the water into the delta area.
4. As the city grew, oil and water pumped from the sediments underlying the city caused the city to sink, in part to sealevel and below. Three of the New Orleans area parishes (counties) are sinking
5. The delta grew in size, to unforseen extents, because of the sediments carried down the river from both natural erosion and the misuse of land that should not have been farmed.
6. As a result of the flood control structures upstream, natural deposits of sediment that counterbalanced the sinking were both diminished and rerouted
6. In addition to receiving the great quantities of sediment, the old river course was altered.
7. As the economic importance of the city grew, it expanded rapidly, far beyond what it had been in the 19th and even early 20th c.
8. There in no unified governing body for metropolitan New Orleans.
10. Louisiana politics is often acrimonious, the northern part at cross-purposes with the south.
11. Construction of ship canals and industrial developments further disrupted the land.

Storm surge from Katrina caused catastrophic failure of the federally designed and built levees, allowing water into 3/4 of the city.

See:
Rivers and Harbors Act of 1965
Flood Control Act of 1965
The Acts, passed by the 89th Congress, authorized the Chief of Engineers (and the Corps) to design and construct navigation, flood control and shore protection in 11 named areas.
Unfortunately the money authorized was small and never renewed to any extent. The projects were numerous, and included:
St. John River Basin, Maine
Atlantic Coastal region, New England
Long Island Sound
etc., etc., to
Lower Mississippi River Basin
Red River Basin
Upper Mississippi River Basin
San Francisco Bay area
Columbia River Basin

A real nice piece of "feel good" legislation. Unfortunately it put people to sleep, was never properly funded, and did not take into account misuse of funds by the federal government and state governments.
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After Hurricane Betsy, control of flood protection was given to the U. S. Corps of Engineers. A project was initiated, to take 13 years to complete. By 2005, when Katrina hit, the project was "between 60-90% complete" (Federal gov't paying 70% and state and parishes paying 30%). See many websites, inc. Wikipedia, for the outlines.
Shortcuts were being taken, the original plan was probably inadequate, and the storm surge, breaching a main ship canal and unfinished areas, brought the catastrophe.