Yes – In some circumstances.
The first thing to do is measure your speed. Use a service like www.speedtest.net or http://www.uswitch.com/broadband/speedtest/. Run the test and record your speed at various points during the day. Take 3 or 4 sample tests.
Next contact your broadband provider and request that they provide an estimate of what speed you should receive based on the service for which you are paying for. The broadband provider should not refuse this information. Refer to Ofcom’s Broadband Voluntary Code of Practice
If there is a mismatch you now have a bit of a slog. The first thing to do is report the problem to your broadband provider. You now need to work with them, frustrating and time consuming as that may be, to identify the root cause of the problems. Part of this investigation may involve a visit to your home by an engineer. You will be advised that if the broadband problem is caused by a problem internal to your home this visit will be chargeable. As such you may want to contact a local telecomms engineer first to ensure that all is fine in your house. (One quick way of checking is to connect your router to the test point on the master socket hence disconnecting your home during the test – see this link for more info on how to do this)
Once you have eliminated your home then you can safely point to the broadband provider as being responsible. They should then make every effort to address the problem and deliver the promised speed.
If problems continue and you do not get the promised service then contact them and ask for a deadlock letter. This should set out what the company has done to try to resolve the problems. You can now take your complaint to the Communications Ombudsman Service and they will investigate. It is at this point you will determine if any compensation is due.
Thankfully the vast majority of cases will be addressed before your get to the Ombudsman stage. Note too that if your home does show as not being the cause of the problem then you should request compensation direct from the broadband provider. Once the problem has been shown to be their responsibility they will be more receptive to providing some form of credit.
Good luck and hope this helps