Teucer was the half-brother of Ajax. Although they shared the same father, King Telamon of Salamis, they had different mother. The mother of Ajax was Telamon's first wife Periboea. The mother of Teucer was a war captive, Hesione, who was the daughter of King Laomedon of Troy.
Since Greeks were fighting the Trojans, being a son of a Trojan war captive not only lowered Teucer’s status but also made his loyalty somewhat suspect. (The madness of Ajax complicates matters in this regard).
Because Teucer was a cousin of Hector, Teucer’s inability to hit Hector with an arrow (actually due to the intervention of Apollo) would be particularly suspect.