Just after 4 pm, I reluctantly tried connecting without going through a splitter and I got connected! Just to prove that it was not due to the splitter, I disconnected and tried to reconnect again using it and was successful. I don't really know what to make of it. I was told it could be due to outside cable problem. It seems there are separate cables for analog (phone) and digital (data).
If this 'on' and 'off' connection problem continues I am thinking of alternative ISP.