Power is in the socket but lamp doesn't light up?

I have two sockets that have power in them.I tested with a tester and power is there. I also tested the junction box to which these sockets are connected and that also has power. If I plug a lamp in either of these sockets, it would not light up. The lamp lights up when connected to other sockets. I think ( I may be wrong) there may be some loose connection somewhere that when the lamp is connected and it draws current, that loose connection can't provide enough power. Any suggestion will be highly appreciated.