I recently had reason to change hosting companies and moved a fair few websites across to the new hosting company.

Hiccups aside, all went well. One question I was asked by a number of customers was "Will this affect my Google ranking?" to which I responded "No. However, if it does it will be positive, not negative."

Interestingly enough, within a couple of weeks 3 sites have jumped in Google. One of them from organic link 10 to about 4/5 (depending when and how you search).

I just wondered if Google really cared about which IP your website sat on, server response times in particular because the move was made to improve the response rates of site pages.

I just wondered if Google really cared about which IP your website sat on, server response times in particular because the move was made to improve the response rates of site pages.

They say it does, but as IPs will resolve to where the hosting company's office is rather than where the server is located i'd apply the pinch of salt.
Same with the oft-quoted "speed factor", it does not say what is used as a comparison for "faster or slower"

AND rankings are of no use as a measure of success.

Not at all.

Changing web hosting won't effect your rankings.