The Bank of England can't run out of money because, if its funds are running low, it can simply print some more. (In practice, the bank doesn't actually print the money. All it has to do is to add a few zeros onto the ends of the amounts shown on its balance sheet. Only 3% of the UK's money supply actually exists as notes or coins. The rest is simply made up of figures on balance sheets).
Such a practice carries high risks because it can lead to massive inflation. Germany tried it in 1923 and prices spiralled out of control. (People had to fill wheelbarrows full of banknotes just to buy a loaf of bread). Robert Mugabe has tried it in Zimbabwe, resulting in prices more than doubling every day.
Despite such risks, the Bank of England has just announced that it's going to follow the same path (using the modern name for the practice, 'quantitative easing'). The idea is that the Bank of England will have more money to lend to other banks, which will then have more money to lend to businesses and individuals. If the theory works, the Bank of England will later start metaphorically burning money (by knocking the extra zeros off the figures in its accounts) before high inflation starts to kick in. The big problem is that such a policy has never been tried before, so getting the timing right will be critical. If the bank claws back the money too early the policy will fail and the UK's economy will, at best, stagnate. If the Bank waits too long, the UK could go rapidly down the same path as Zimbabwe.
Chris