You can trade under any name you like but no-one else could claim to be 'yourcompany limited' once you have registered it.
A limited company can trade under any name it likes also, for instance 'The Boots Company PLC' trades as 'Boots' our high street chemists.
I found this on the net
There is an important distinction between the official title of the company and the name under which it may trade.
It is possible to for a company to be registered under ABC Limited and then start trading as the business name XYZ . The main difference between the two is that the name ABC limited is regulated by Companies House, and as such, only one such company may be set-up with it.
XYZ however, belongs to the realms of trademark legislation and business names, which are generally far more difficult to monopolise. In order to register a trademark, the business must demonstrate that the name is firmly associated with them in the eyes of the public or other group relevant to their activities.
Thus, even if a company has incorporated using XYZ limited, this would probably not prevent other businesses from trading as that name. There might be legal ramifications should it be judged that one business is passing itself off as the other in order to gain trade unfairly.
The Companies Act 2006 introduced new provisions which came in to effect on January 1st 2007 requiring all websites to display certain additional details. This extra information includes the name, company number and VAT registration code of the corporate owner of the website.
Although these amendments did not diminish the capacity of companies to trade as another business name, it does increase the transparency of the relationship and thus, might result in less confusion as to who the visitor is engaged with.
My company is AFSystems.co.uk Limited and I trade as 'AF Systems' but I also trade as 'Carpet & Upholstery Care' and 'Stafford Window Cleaning' there is nothing illegal about this as long as I declare that these companies are all the same.
You could have 2 people living next door to each other trading as window cleaners both trading as 'Bob the windie' and there is nothing either could do about it. It may be a little confusing for the customer though!