Brett Favre had the most games when he had 14 or more (ATT/GP) and with 1+ intercepted pass, with 196 games.

most games with 1 interception, mimimum 14 pass attempts per game
Interpreted as:
most regular-season games by a player with 1 or more passing interception, mimimum 14 or more pass attempts per game
FavreBreesMarinoManningTarkentonTestaverdeManningElwayUnitasHadlRiversMoonHartRoethlisbergerBradySneadFoutsBlandaKriegBrodieFergusonTittlePalmer196157157154153150147145144143136136136134133131129123120120119119118
NAMEGPATTINTATT/GPCMPPCTYDSAVGYDS/GTDTD%INT%SCKSCKYRATE
Brett FavreBrett Favre1966,90033635.24,13359.946,2426.7235.92874.24.93522,40273.5
Drew BreesDrew Brees1576,13224339.063,95764.544,3767.2282.62764.54.02621,94584.5
Dan MarinoDan Marino1575,63625235.93,28258.241,1437.3262.12815.04.51821,31979.0
Peyton ManningPeyton Manning1545,78625137.573,67263.542,4877.3275.92915.04.31961,26884.3
Fran TarkentonFran Tarkenton1534,26526627.882,38255.830,1347.1197.02014.76.267.8
Vinny TestaverdeVinny Testaverde1504,70026731.332,61955.731,9696.8213.11974.25.72751,99767.2
Eli ManningEli Manning1475,29524436.023,12158.936,3246.9247.12184.14.62711,93974.3
John ElwayJohn Elway1454,70322632.432,60655.432,1116.8221.51753.74.83312,50369.1
Johnny UnitasJohnny Unitas1443,84525326.72,07754.028,7547.5199.71955.16.667.7
John HadlJohn Hadl1433,57126824.971,74548.925,0177.0174.91704.87.556.6
Philip RiversPhilip Rivers1364,89220935.973,08163.036,2097.4266.22274.64.32971,80183.1
Warren MoonWarren Moon1364,67223334.352,69557.733,3147.1245.01864.05.03172,34172.4
Jim HartJim Hart1363,86824728.441,93350.025,9146.7190.51493.96.457.9
Ben RoethlisbergerBen Roethlisberger1344,73620135.342,98263.035,0667.4261.71944.14.23152,12581.4
Tom BradyTom Brady1334,92919137.063,07662.435,1507.1264.32264.63.92441,51582.9
Norm SneadNorm Snead1313,45625726.381,77151.224,0247.0183.41464.27.456.8
Dan FoutsDan Fouts1294,22824232.782,44557.831,9767.6247.91774.25.771.9
George BlandaGeorge Blanda1233,28027226.671,56747.822,1316.7179.91875.78.354.5
Dave KriegDave Krieg1203,58219929.852,03756.924,6716.9205.61554.35.63072,39169.4
John BrodieJohn Brodie1203,28322427.361,76953.921,7436.6181.21344.16.859.8
Joe FergusonJoe Ferguson1193,19520926.851,64651.520,7756.5174.61223.86.557.6
Y.A. TittleY.A. Tittle1192,82522123.741,52854.120,3847.2171.31435.17.861.5
Carson PalmerCarson Palmer1184,17718735.42,56461.430,2597.2256.42024.84.52241,57980.9
Kerry CollinsKerry Collins1184,01819634.052,17854.225,5846.4216.81263.14.92161,43063.9
Drew BledsoeDrew Bledsoe1174,21120635.992,33055.326,9536.4230.41293.14.93062,06864.7
StatMuse has game-level data for intercepted passes and passing attempts going back to the 1950 season.

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