Can you run a 4k monitor at 30hz with a late 2011 MBA? Is it possible to run an external display connected to a late 2011 MacBook Air at 3840x2160@30Hz. 

I realize having this resolution at 60hz exceeds the bandwidth limitation of a Thunderbolt 1 port. This is why I am asking specifically about 30hz.

Hardware: Late 2011 MBA running 10.9.3

Monitor: Samsung U28D590D, connected to MBA with Mini DP to DP cable

This isn't a duplicate question. The other related asked-and-answered questions all point out that this impossible for 4k@60hz. I am inquiring about 3840x2160@30hz specifically.
No, this is not possible. As Apple states, the only models capable of supporting 4k resolutions (3850x2160 UHD) at 30 or 60 Hzs are the late 2013 Mac Pros and late 2013 Macbook Pros:


  "You can use 4K displays and Ultra HD TVs with your MacBook Pro
  (Retina, Late 2013) or Mac Pro (Late 2013) computer. You can use these
  displays at the following resolutions and refresh rates with the
  built-in HDMI port 
  
  3840 x 2160 at 30 Hz refresh rate"


In fact, this is the default configuration. 

Though these two models also are capable of supporting 4k @ 60 Hz refresh, multi stream transport must be enabled on the external display. 

The same applies to your Thunderbolt port. Thunderbolt 1 is not capable of driving 4K displays. Thunderbolt 2 is.