Why are my Shimano 105 11-speed shifter/derailleur acting like they're 10-speed? I recently got a pair of Shimano 105 shifters and a rear derailleur and put them on my bike. The front derailleur is a Shimano Tiagra, and the cassette is also Shimano. The cables are new, but the outer cables are old.

I managed to get the rear mech indexed correctly when I installed them, but the following day the bike wasn't shifting to the biggest sprocket. I tried to adjust the indexing, but if I loosen the barrel adjuster enough to shift to the biggest sprocket, the derailleur won't shift to the smallest one. Similarly, I tighten the adjuster until it shifts to the smallest sprocket, it won't shift to the biggest one.

If I pull on the cable and turn the cranks with my other hand, the derailleur shifts to the biggest sprocket but the chain drops back to the second biggest when I let go of the cable.

I have looked at the invoice and the packaging, and it says there the components are supposed to be 2x11, and I counted ten clicks from the shifter. What could be the cause of this issue?

Here are a few photos of the rear mech:

Rear view:



Now that I look at it, the derailleur cage seems to be in an odd angle. Here's the side view:
It's evident from your first picture that there is definitely a pretty bad derailleur cage alignment issue. 

The derailleur hanger does not appear bent, but it does look like the derailleur bolt may be cross threaded , as the derailleur body seems to be at an angle to the hanger.

Check the derailleur installation and also check that the cage is not bent.