^
There is no reason why you wouldn't technically be able to do it. I've never heard of anyone who has done a masters straight after the LPC though.
I'm not sure if it is sensible though- the LPC is fairly practical, and is VERY VERY different from uni or the GDL- it doesn't really make sense to do this, and then go back to academia with a masters.
There would also be serious money issues. You need to pay the fees for the LPC and for accomodation etc; it is more expensive than uni.
Most people try to secure a training contract before doing the LPC, when you secure a training contract with a firm, the firm generally pays your fees and gives you a couple of grand maintenance. If you then went off and did a masters without doing the TC, you would have to pay that money back. A lot of people also apply for jobs during the LPC, again, you often get the fees paid if you are successful in doing this, but won't be able to do this if you go and do a masters straight after. So you are in a much better position financially if you go and do a masters and then do the LPC.