As far as I'm aware, the name usually comes from the number of bells in the local church tower. Six seems to be a common number. (Campanologists might be able to offer an explanation for that). But there are 'Five Bells' pubs all over the country, e.g.:
http://www.fivebells.net/
There are quite a few 'Four Bells' pubs around as well
(e.g.:
http://www.diningpubs.co.uk/pub_details.asp?id =195 )
but perhaps the name is less common since only the inn-keepers in parishes with an unusually large number of bells (in the tower of the parish church) considered it worth proclaiming in the name of their inn.
Higher numbers of bells are likely to be less common, simply because most church towers would be unable to take the weight of all the bells. However, just up the road from me, there's this pub:
http://www.stonhamaspal.co.uk/inf/tenbells/ind ex.htm
Chris