In which cases do KCL and KVL fail to apply in circuits? For which circuits will KCL and KVL be applicable?
 A: The assumptions of circuit theory are as follows (see Nilsson and Riedel, Electric Circuits, ch 1):

*

*there is no net charge on any circuit element


*there is no magnetic coupling between circuit elements


*the circuit is small compared to the speed of light and the time scales of interest
KCL and KVL can be used for all circuits that satisfy these three assumptions. They fail in circuits that violate these assumptions.
A: Item #2 in @Dale's answer is specialized to two-terminal circuit elements: it excludes transformers, for example. To include those you must allow four-terminal elements (2-ports) to represent magnetic coupling. You can also allow and include 3-terminal elements, such as transistors, gyrators, multi-terminal capacitors, etc. These can be defined by their terminal (port) current/voltage relationships and thus embed them in a network so that all loops and nodes follow KCL and KVL.
A: I'll focus on KVL. It makes sense to distinguish between "KVL is true" and "KVL is applicable".
KVL (shorthand for Kirchhoff's voltage law) is a useful engineering rule inspired by the original Kirchhoff's second circuital law. The latter states that sum of terms $R_k I_k$ over all elements in a closed path is equal to sum of all electromotive forces acting in the closed path. This is essentially a generalization of Ohm's law, and requires that the various members of the closed path can be assigned ohmic resistances $R_k$ that do not change with current. This original law is approximate and requires ohmic behaviour from the underlying conducting paths, which is well obeyed by metals in usual ranges of current and temperatures, but not by bad or unusual conductors.
The modern KVL in terms of voltages, however, is always true, because instead of emfs, it refers to voltages defined as difference of electric potential. It is a mathematical fact that sum of differences of any scalar field when completing a closed path must be zero. This is often very useful in formulating equations for basic circuit models with lumped elements.
However, this KVL, while it is always true, is not always directly applicable to all circuits, because it may not be possible to express difference of potential on an element in terms of other quantities describing that circuit, like currents.
A very important example is the circuit that contains one winding of a transformer: voltage on the terminals of the transformer winding isn't determined by variables describing that circuit only; instead it is influenced by details in the other winding, even though it is not part of the circuit. This is because the other winding induces additional electromotive force in the first winding, which breaks the simple relation between induced emf and difference of potential valid for isolated (uncoupled) inductors.
A simpler example: we have two resistors, $R_1 = 1\Omega$ and $R_2 = 1000 \Omega$, connected to each other so they form a circuit, and there is induced emf $\mathscr{E} = 1001 \text{V}$ in the circuit due to magnetic flux changing in time. What is the current in the circuit?
KVL, while true and valid, is not directly applicable in a practical sense to this problem, because difference of potential on any of the resistors is not determined by current flowing through that resistor or some other circuit variable. Potential differences are not determined, as they depend on details of charge distribution on the wires and inside the resistors, not merely on current $I$ flowing in the circuit. Potential differences can be even zero on both resistors, despite net emf and current being nonzero.
More generally, in cases where difference of potentials on a conductive member (resistor, coil, ...) cannot be determined by known electromotive forces, resistances, current and its derivatives, KVL cannot be usefully applied. However, sometimes the original Kirchhoff's second circuital law can be applied instead.
In our example with two resistors, we get
$$
R_1 I + R_2 I = \mathscr{E}
$$
which implies that current $I = 1 \text{A}$. Potential differences on resistors remain undetermined.
