Can I get a 1 year UK long term visitor visa?
It's a moot point because even if they did offer such a visa you would not qualify. The longer term visas are for people with a lengthy performance history and who have a really strong premise.
You want to cohab with your boyfriend while he attends graduate school in the UK, depending on your cohab history you might be able to qualify as a trailing partner. You would need some very hard evidence of two years continuous cohab and evidence demonstrating a fully partnered lifestyle, and these visas (unmarried partners) are hard to get in the first instance. To investigate this, your boyfriend's school has an officer entitled, "Authorising Officer" who can provide more details.
That leaves you with the visitor programme, and while the Immigration Officer who lands you can authorise up to six months, there's no guarantee that she actually will. Six months is the maximum and generally issued, but it's not a bankable assumption, especially if you show up without credible onward travel arrangements within the six month window and some hard accommodation evidence.
The implication here is that you will 'channel hop' when your expiry date approaches and then reenter. People do this with varying measures of success. The IO will see that you are channel hopping and wonder what sort of lifestyle you have in the US that allows you to be gone for so long and what you are doing to support yourself in the UK (hopefully nothing illegal). For these reasons, the landing interview is more probing, more intense, and hence more risky. If you are unlucky and get removed from port, you will have a history that could prove awkward or embarrassing in later life (in addition to restricted mobility as far as the UK and Europe go).
The final option, as explained in the comment by 'phoog' is to qualify for a student visa that runs parallel to your boyfriend's visa. Questions about this should be taken up on Expats. They also deal with trailing partner visas.