Assuming you shop smartly and know what you are buying...
A $500 bolt rifle can be pretty accurate. 1 MOA out of the box is quite do-able. A $500 semi-auto 308 does not exist.
A $1000 bolt rifle can be even more accurate, but you have to make sure you are buying accuracy/trigger and not a nicer stock or a more fashionable label. A $1000 semi-auto might exist (I am not aware of any), but it won't be be competitive with the similarly priced bolt.
At about $1600 you start to see good, accurate semi-autos like the M1A, AR-10 platforms, FN-FNAR. These are usually 1 MOA, more or less. But for similar money, you could get much more accuracy out of a bolt gun.
At the higher priced levels ($3k and up), with lots of customization, I don't really know what is possible... not my area of expertise.