build up or buy?

Are there any significant savings to be had in building up a bike versus
purchasing a complete bike at the LBS? I've started saving up for a new
bike but thought about the prospect of slowly building a bike up over a
time period. While I think it would be a worthwhile endeavor to piece
together a bike, I don't know if it would save any money or if it might
actually be more expensive.

Thanks,
chris