What is the basis for the belief by some groups that a Christian can live a sinless life? I understand that there are some Christian groups that believe that it is possible for a Christian to live a life entirely free from sin.

What is the nature of this belief and what is the basis for it?
I think you are referring to a doctrine called Entire Sanctification (also Sinless Perfection), which is taught by some Wesleyan and holiness churches.  A Google search on that term will give you more information, but basically, the idea is that it is possible with the Holy Spirit's help to avoid sin completely.  Some Scriptural basis for this belief can be found in Romans 6, 1 Thessalonians 5:23, and 1 John 2:1-2, which talk about Christians being free from sin and blameless.