Bradley and Patton were never friends or knew each other before WW II but they got along fine during the war because both hid their real feelings.
As shown by his diaries, Patton thought he was better man and general than Bradley. To his face, however, Patton was the loyal subordinate. The same is true of Bradley. He never criticised Patton during WWII for conduct of the Sicily campaign.
Its only after Patton's death and his publication of his papers/diaries that Bradley went public with his feelings.
As for true feelings. Bradley thought Patton while willing to take risks, great at pursuit, and a "thruster" was a sloppy administrator and adverse to detailed planning. Further, he thought Patton was too interested in public publicity. Patton, OTOH, thought Bradley had no vision or imagination, feel for the enemy, and was unwilling to take risks.
IMO, the men worked best when Patton was boss. Bradley could temper Patton recklessness, and was able to perform the detailed planning necessary to implement his ideas.
Further, Bradley was an example of the peter principle. His natural ceiling in my opinion was Army Commander. He had no ability to put himself in the enemies place or understand what the Germans could do or couldn't do. (Despite ULTRA!) For example:
1) Prior to D-day he thought that the landings would be easy, the tough part would be stopping the enemy counterattack ala Salerno or Anzio. Completely wrong.
2) He completely missed the significance of the Hedgerows on military operations.
3) After the breakout he constantly reined Patton in, and forced him to provide flank guards, completely misreading the ability of the Germans to counterattack. The failure to close the gap at Falsaise is his fault.
4) Having overestimated the Germans during the Normandy breakout, he went the other extreme and considered the war won in late August and early September. Advancing on a broad front, he threw away any chance of reaching the Rhine.
5) He continued to underestimate the Germans throughout Sept-Dec 1994. Launching penny packet attacks all along the front, he incurred thousands of causalities while accomplishing nothing. He was taken surprise during the Battle of the Bulge because he thought the German could never counterattack.
6) After the Bulge, he went back to Overestimating the Germans. Demanding that the ENTIRE West bank of the Rhine be occupied before any further advance. Holding up Patton. Demanding the Ruhr pocket be eliminated before any further advance into Germany, etc.
7) Finally, it should be noted that Bradley was against Patton's short-hook landings in Sicily, even though this was correct strategy and hastened the Germans withdrawal. The only thing wrong with them, if fact, was that weren't done sooner and in bigger strength. But Bradley didn't like to take risks.