First, the context. This phrase is found in a paragraph about causing the church to grow into christlike maturity (4:13-17). The result of this maturity is that the believers are no longer deceived by every teaching that comes along (4:14). In contrast, we are to "speak the truth" and thus continue to grow to be like Christ (4:15-17). So what Paul is talking about here is a contrast between a church that accepts all sorts of false teaching and one that speaks the truth. In other words, Paul is talking about teaching the Truth out of love so that Christians will become more like Jesus.
This interpretation, that Paul is talking about right teaching, is verified by the meaning of ἀληθεύοντες, "speaking the truth" (one word in Greek, not three). Paul uses the word elsewhere only in Gal 4:16: "Have I now become your enemy by telling you the truth?" Here, as in Ephesians, "telling the truth" is about teaching the truth.
The NET Bible translates this phrase as "practicing the truth" because the verb is sometimes used in the LXX to imply being true or proving true. However, in determining the meanings of words, we should let near context rule over far context. Eph 4:13-15 contrasts ἀληθεύοντες with false teaching, not with unfaithfulness; and Paul's use of the word in Gal 4:16 is about teaching the truth. It is difficult to follow Paul's train of thought in Eph 4 if he is talking about being faithful or true rather than teaching the truth.
The picture: Eph 4:15 from Codex Sinaiticus, a manuscript from AD 350. The first word in the first line is ἀληθεύοντες, speaking the truth.