Interest rates are used as a tool for managing inflation and the economy as a whole. Basic economics (ie, supply and demand). If the economy starts to overheat, interest rates are increased and continue up until the economy slows down and then they start going the other way (slowly) in order to prevent pushing the economy into recession. Timing is crucial and I suspect any Government is a poor judge, hence probably why the BoE was made independent of Government when Labour came to power. It may be coincidence that the UK has managed to have the longest sustained length of growth in recent history with a recent gentle drift downwards.
Bank and building societies set their interest rates on a commercial decision based on their own cost base. The more expensive the rate then probably the more expensive (and perhaps even inefficient) the lending organisation. You usually find the older, longer established financial services organisations charge higher interest rates for loans, mortgages, etc. whereas the new kids on the block (ie, internet based banks), can offer low rated borrowing rates because there costs are lower (or they're just more efficient). It works in reverse for savings rates.
I wouldn't think any bank could sustain indefinitely to match the BoEs base rate for borrowings or savings because they wouldn't generate any margin to meet their overheads.