Line of Sight (LOS) and High-Gain Antennas

Will a high gain antenna help transmit through obstacles if the radio path is non-line-of-sight (non-LOS)? 

When speaking about radios and line-of-sight, it must be understood that visual LOS and RF LOS are two very different concepts. RF LOS refers to a clear Fresnel zone between two radio antennas while visual LOS only refers to a being able to see the other object in a straight line.

To achieve best results, Digi recommends using only medium gain antennas in non-LOS conditions. The very high gain antennas actually can perform worse in non-LOS conditions because they have a narrower beam width. The multipath effects of a non-LOS environment cause RF signals arrive at the antenna at odd angles after reflecting off of nearby objects. A lower gain antenna with a larger beam width may be better at recombining the bounced signals.

Generally speaking, a 6dB improvement in the link budget will double the range in a LOS environment (8dBi gain antenna compared to a 21 dBi gain di-pole antenna). To double the range in non-line of sight conditions it takes at least 12 dB of additional receive sensitivity or transmit power.
However, 12 dB of additional antenna gain may not double the range because of the narrower antenna beamwidth which causes the antenna to ignore signals arriving from angles outside of the beamwidth. Improving receive sensitivity is generally the best option to improve range.  On some Digi radios, this can be done by decreasing the over-the-air baud rate.