Ben Roethlisberger has the most games with 501+ passing yards, with 3 games.

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who has the most career 501 passing yard games
Interpreted as:
Who has the most career 501 or more passing yard regular-season games?
RoethlisbergerBreesPrescottGoffCarrRyanRiversRomoSchaubManningStaffordBradyGrbacEsiasonMoonMarinoSimmsFerragamoTittleVan Brocklin32111111111111111111
NAMEGPYDSCMPATTPCTAVGYDS/GTDTD%INTINT%SCKSCKYRATE
Ben RoethlisbergerBen Roethlisberger31,53111316170.29.5510.3116.800.0851123.0
Drew BreesDrew Brees21,0157610274.510.0507.598.854.926114.6
Dak PrescottDak Prescott1502415870.78.7502.046.911.7321112.9
Jared GoffJared Goff1517456866.27.6517.022.934.422780.3
Derek CarrDerek Carr1513405967.88.7513.046.800.0215117.4
Matt RyanMatt Ryan1503283775.713.6503.0410.812.7322142.0
Philip RiversPhilip Rivers1503436566.27.7503.023.100.031599.7
Tony RomoTony Romo1506253669.414.1506.0513.912.8436140.0
Matt SchaubMatt Schaub1527435578.29.6527.059.123.6223121.7
Eli ManningEli Manning1510315160.810.0510.035.935.90089.5
Matthew StaffordMatthew Stafford1520365961.08.8520.058.523.4218103.8
Tom BradyTom Brady1517324866.710.8517.048.312.111121.6
Elvis GrbacElvis Grbac1504395373.69.5504.023.823.843099.9
Boomer EsiasonBoomer Esiason1522355959.38.8522.035.146.821577.1
Warren MoonWarren Moon1527274560.011.7527.036.700.0322123.1
Dan MarinoDan Marino1521356058.38.7521.035.058.30068.8
Phil SimmsPhil Simms1513406264.58.3513.011.623.277082.3
Vince FerragamoVince Ferragamo1509304665.211.1509.036.524.313106.2
Y.A. TittleY.A. Tittle1505273969.212.9505.0717.900.0151.4
Norm Van BrocklinNorm Van Brocklin1554274165.913.5554.0512.224.9128.3
StatMuse has game-level data for passing yards going back to the 1950 season.