Is the Nether infinite?
From Minecraft Wiki's page on The Nether:
... Under the Lava ocean, there is a layer of Nettherack1 and Bedrock on the very bottom. Similarly, the very top of the Nether is Bedrock.
1 Or Bloodstone
Basically, going up or down either way leads to Bedrock.
The Nether is as infinite as the surface horizontally, however, like the surface it is limited to 128 blocks in height. The only difference is that in the Nether there is bedrock above and below, whereas the surface only has it below.
To be technical, neither the Nether nor the surface are infinite (both however are VERY VERY VERY LARGE). Regardless of horizontal limits however, ther vertical limit is 128 blocks, just as the surface is.