NEX-6 closing the shutter after 1 picture.

This may be a silly question, but is the battery fully charged?

 

Sounds like it. Sad, but every line from every manufacturer has the occassional lemon.

 

Bummer on the first one, but glad they got you a replacement!

 

Good news. I was reluctant to offer this up before, but I will now.

The thirty-six inch drop test. This was suggested to me by a USMC radio technician while we were in the wilds of South Korea with a malfunctioning aircraft radio.

Whenever a piece of electronic equipment fails to work as advertised and all standard tests have failed to resolve the issue we recommend the 36" drop test.

Step 1) Find a nice flat hard surface. Concrete works well.
Step 2) Hold the offending piece of electronic equipment over said hard surface at waist level.
Step 3) Release the offending piece of malfunctioning electronics equipment allowing gravity to work its magic.
Step 4) After impact pick up offending piece of electronic equipment and attempt to operate normally.
Step 5) If offending piece of electronic equipment now functions- Congratulations! You fixed it!
Step 6) If offending piece of electronic equipment still does not function properly- Oh well, it didn't work before anyway.

While I've never reached Step 5 personally, there is a certain satisfaction to be garnered by performing Steps 1-3. YMMV