The Markovian assumption is used to model a number of different phenomena.
It basically says that the probability of a state is independent of its history, but only depends upon its immediately previous state.
That is,
$p(x_i|x_0,x_1, ..., x_{i-1}) = p(x_i|x_{i-1})$
Such a property has been used to model a number of phenomena, like the strength degradation of a structural component using dynamic bayesian networks.
I'm curious, what could be the limitations of such an assumption? Can you give some practical examples when such an assumption would be invalid? What could be the consequences in such a case?
Any situation where a system has hidden state is going to be non-Markovian. For this, people invented hidden Markov models, where the system is assumed to be Markovian with a larger state but we can only observe part of it (see for instance the concrete example given here). Every system can be written as a hidden Markov model in a boring way: if the sequence is $X_1,X_2,\ldots$ then we can consider it as hidden Markov with states being sequences themselves $(X_1,\ldots,X_n)$ transitions to $(X_1,\ldots,X_{n+1})$ and the hidden Markov function maps a sequence to its final element.