'Is this possible that beer could be brewed up somehow with weed?'
Sure, in the same way you can put any sort of anything in your beers, as long as you think it will taste good. But to answer the more important question:
'[W]ould you get the effects of both beer and pot both?'
It would very much depend on how you used it in the process.
THC, the psychoactive chemical in marijuana, is barely soluble in water, 0.0028 mg/mL or 2.8 parts per million (according to Wikipedia). So if you added your herb to the boil, that's the most you could hope to get (a 20 liter batch of beer could be THC-saturated with just 0.6g of 10%-THC pot).
If you added it as a 'dry-herb' addition you would see a much higher solubility due to the presence of the alcohol formed during fermentation. THC is soluble in pure ethanol at 35 mg/mL (or 35,000 PPM)(source). Say your beer is 5% ABV and that's ~1,800 PPM it could contain. If my math is correct, that would take roughly 360g of 10%-TCH pot to fully saturate 20 liters, just about 9g. worth of potency you could hypothetically pack into each pint.
One key to this process you don't want to overlook (as noted in these instructions for making a tincture of marijuana) is that the psychoactivity of cannabis is dependent on heat-activaction of inert THCA to become active THC (when you smoke, vaporize, or bake the pot, this heat-activation is already part of the process). As outlined in the link, the herb must first be heated to cause this chemical change, but not so hot that it vaporizes the THC (225°F is recommended). If you skip this step before adding the herb, even if you are able to dissolve its chemicals, they may not be the ones you're after.
Since you'd have to add the herb post-fermentation anyway, this seems like the perfect kind of thing to experiment with using single bottles of beer, dosing and aging differently to see what works.
Here's a thread on Reddit in which someone with some experience goes into more detail on the process.
And an older recipe from Erowid for marijuana beer.
Both mention another feasible way to add the pot to the beer, which I didn't consider. Basically you add it to the boil, as you would hops, and you make sure to transfer the entire contents of the kettle to the fermenter (hops, trub, everything you would normally leave behind) so that the THC-bearing plant material is in the fermenting beer as alcohol is produced.