The cap is only for certain kinds of councils. Many people also pay a precept to a lower level council (eg, town or district) which can be raised as much as they like. Different counties have completely different council structures so in different areas different levels of government pay for different things.
So if you're thinking - that can't be right, it might be for another county!
Anyway, I live in Northants.
We're having a fun time at the moment... massive council tax rises and huge cuts to everyday services. They have acknowledged that their budget plan is to maintain roads at a less than legal level even though it will leave them vulnerable to court cases. The government inspection report says a lot about central government attitudes to struggling councils (ie, tough shit cut to the bone and abandon any ideas you previously had about what was the bare minimum you wanted)
In Northants obviously our council has obviously made massive mistakes, but it's also had huge reductions in funding from central government.
We have mostly Tory Councillors (at all levels) and all Tory MPs. They are all standing around shouting at each other about whose fault it is and has been for months.
Similar problems going on in Labour councils, but at least they don't have the added dynamic that NCC has, where they were told by the Conservative party not to raise council tax for years for image reasons, when they really should have for planning reasons - and now they are being shafted by the party that told them to do that.
MPs and Councillors have all proved they're shockingly bad with money...
I don't usually say this, because voting is so personal but anyone voting Conservative here next time around needs their head checking.