Hurricanes Katrina, Rita, and Gustav were relatively dry storms. Had they been wet storms like Allison, the damage to Louisiana would have been much greater. If New Orleans received the 39 inches of rain that Houston received during Allison, it would overwhelm the pumps and would flood nearly as badly as with Katrina.
https://biotech.law.lsu.edu/climate/docs/allison.pdf
“Tropical Storm Allison produced severe storms, torrential rainfall, and associated flooding across the southern and eastern sections of the United States from June 5 to June 18, 2001. After making landfall near Galveston, Texas, on June 5, the storm moved inland to near Lufkin, Texas. Allison drifted back into the Gulf of Mexico on June 9, turned to the northeast, and made landfall again on June 10 near Morgan City, Louisiana. After causing 24 deaths in Texas and Louisiana, Allison moved across southern Mississippi, southern Alabama, southwest Georgia, and northern Florida, causing 9 more deaths. By mid-week, Allison stalled over North Carolina and produced more heavy rainfall and flooding before tracking northeast along the DelMarVa Peninsula and moving off the New England coast on June 18. Seven additional deaths occurred in Pennsylvania and one in Virginia. Figure 1 shows the path of Tropical Storm Allison, and Figure 2 shows the associated rainfall. Tropical Storm Allison caused more damage than any tropical storm in U.S. history, with estimates in excess of $5 billion. Most of the damage and fatalities (22) occurred in Houston, Texas. Storm rainfall totals peaked at 36.99 inches (Port of Houston) in Texas and 29.86 inches (Thibodaux) in Louisiana. Since this was the area of extreme rainfall and greatest impact in terms of damage and fatalities, the report focuses on NWS performance in southeast Texas and southern Louisiana.”