I've very often seen it come up as lack of communication, either resulting from or leading to heartache.
Heartbreak is another possible meaning, especially when a third person ("the other woman/man") is involved.
Feeling left out, self-pity, jealousy, envy.
I'll give you two personal examples, maybe that will help:
1) One morning I drew the 3 of Swords as my card for the day (along with the 9 of Swords and 10 of Swords). I had been having issues with my best friend: I was jealous of the time he devoted to his new girlfriend. I had been feeling left out and lonely, but I couldn't tell him that, because I knew I was being unreasonable and selfish. In the end we had an argument where I still couldn't say what I had to say. This lack of communication meant we kept being angry and hurt. I also learnt that he had been keeping important health-related concerns from me, and I felt betrayed. We did eventually resolve our differences but this was truly a 3 of Swords moment for me: jealousy, feeling left out, heartache, betrayal.
2) After my ex broke up with me, I kept drawing the 3 of Swords. I was in complete denial of my grief; unsurprisingly, it turned out to mean that I had to acknowledge my heart was broken in order to heal.
Hope that helps!