How limiting/ heavy is the "triangle inequality" assumption? Suppose a theorem proves something about a family of distance measures, with this the triangle inequality assumption. How limiting this assumption is in reality?
What are some real-world examples of distance measures which don't obey the triangle inequality? 

Edit:
The triangle inequality is the following requirement upon the distance function on $M$:
$$
\operatorname{dist}(x,z) \leq \operatorname{dist}(x,y) + \operatorname{dist}(y,z), \qquad \forall x,y,z \in M.
$$

A real-life "distance" with a weakened triangle inequality is the round-trip delay time.
EDIT: the RTT is the time it takes for a signal to make the journey back and forth between two nodes on a network. According to The Inframetric Model for the Internet:

A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the round-trip delay (RTT) is a metric.

But...

... we confirm that the triangle inequality does not hold for a significant fraction of the nodes. Nevertheless, we demonstrate that RTT measures
satisfy a weak version of the triangle inequality: there exists a small constant $\rho$ such that for any triple  $u$, $v$, $w$, we have $RTT(u,v)\le\rho\max\{RTT(u,w),RTT(w,v)\}$.

