Prior to 1930, the three dominant luxury car manufacturers in the US were Packard, Peerless and Pierce Arrow. All three out sold Cadillac and dominated the much more limited production by companies like Dusenburg, Stutz, Brewster and Locomoble. General Motors tried to enter the high end luxury car market with Cadillac, but it wasn't until they introduced the V-16 engine that Cadillac really got a toe hold in this market.
The Great Depression also helped. Peerless had been declining during the late 1920s and went bankrupt in 1932. Pierce Arrow was bought out by Studebaker in 1928 and was in dire straights before Pierce Arrow was able to regain its independence. Pierce Arrow never recovered and finally folded in 1938. Packard was able to ride out the Depression by introducing lower cost models, like the 120. Packard was still the leader in the high end luxury car market and was selling three times as many cars as Cadillac as late as 1948.
However, many of these Packards were lower cost cars, competing in the Oldsmobile, Mercury and Chrysler price range. That devalued the Packard mystique. Harley Earl also began to style better looking Cadillacs, introducing such innovations as tail fins and automatic transmissions. GM was also able to retool the Cadillac production line every year, where as Packard could only change their body styles every three years. When Chrysler bought Briggs in 1954, Packard lost its provider of body shells and had to develop their own in house source for body work. Quality declined drastically after 1954 as their fit and finish was certainly below the standards of a premium luxury car. A merger with Studebaker did not help either. By 1950 Cadillac became the leader in the luxury car market segment. Except for a short period around 2000, when Lincoln briefly out sold it, Cadillac has been the premiere American luxury car brand.
Sources: "Fifties Flashback : The American Car" by Dennis Adler
"Packard" by Dennis Adler
"More than they Promised: The Studebaker Story" by Thomas Bonsall
"The Complete History of General Motors: 1908-1986" by Richard Langworth and Jan Norbye