You can filter the data within geom_text to keep only the row with the highest MOB value within each Vintage . The plot is still pretty cramped though:

colors = c("#e6194b", "#3cb44b", "#ffe119", "#4363d8", "#f58231", 
           "#911eb4", "#46f0f0", "#f032e6", "#bcf60c", "#fabebe", 
           "#008080", "#e6beff", "#9a6324", "#fffac8")
ggplot(data=curves, aes(x=MOB, y = NCO_Rate, colour = Vintage)) +
  geom_line(aes(), size=1) +
  labs(title = "NA") +
  theme_classic() +
  scale_color_manual(values=colors) +
  geom_text(data=. %>% 
              arrange(desc(MOB)) %>% 
              group_by(Vintage) %>% 
              slice(1), 
            aes(label=NCO_Rate), 
            position=position_nudge(0.1), hjust=0, show.legend=FALSE)
Rplot766×538 49.5 KB