Being widely accepted in psychology has a great meaning. In the example of the Big Five Inventory (BFI), the inventory has a very strong number of studies backing its validity and reliability. There is a reason why it's widely accepted. Gosling, Rentfrow, and Swann Jr (2003) conducted two studies investigating several measures of validity and consistency. The authors found when comparing BFI to other well-known personality tests.
High convergent validity (r = .77) - items measuring a specific construct were strongly correlated with items from a different inventory measuring a similar construct
Low discriminant validity (r = .20) - items measuring a specific construct were not strongly correlated with items from a different inventory measuring a different construct
Higher test-retest reliaiblity (r = .80) - measuring a group of participants yielded highly similar results on different measurement occassions
Higher prediction of external correlates - BFI yielded stronger correlations to specific measures than other tests (e.g., physical attractiveness)
There is more data if you are interested.
Source: http://anthro.vancouver.wsu.edu/media/Course_files/anth-260-edward-h-hagen/tipi.pdf