The derivation of Fermi's golden rule uses the following approximations:
Transition time is small.
Photon frequency $\omega\approx\omega_f-\omega_i$.
The question is, are there any experiments where Fermi's golden rule cannot be used because either one or both approximations are not valid?
EDIT:
Just to clarify, this question is about the derivation of Fermi's golden rule, not the rule itself. Please see the following video series to learn more about the derivation of Fermi's golden rule and the approximations that are used.
https://www.youtube.com/watch?v=fTLTSnqVnNA
My notes from the video series and Quantum Mechanics for Scientists and Engineers.
Step 1. Define $\Psi(x,t)$ for $\psi_1(x)$ the initial eigenstate, $\psi_2(x)$ the final eigenstate. \begin{equation*} \Psi(x,t)=c_1(t)\psi_1(x)\exp(-i\omega_1t)+c_2(t)\psi_2(x)\exp(-i\omega_2t) \end{equation*}
Step 2. Define the perturbing Hamiltonian. \begin{equation*} H'(x,t)=2V(x)\cos(\omega t+\phi) \end{equation*}
Step 3. From time dependent Schrodinger equation \begin{equation*} i\hbar\frac{\partial}{\partial t}\Psi(x,t)=H_0(x)\Psi(x,t)+H'(x,t)\Psi(x,t) \end{equation*}
Step 4. Result \begin{equation*} i\hbar\frac{\partial c_1(t)}{\partial t}\psi_1(x)\exp(-i\omega_1t) +i\hbar\frac{\partial c_2(t)}{\partial t}\psi_2(x)\exp(-i\omega_2t) =H'(x,t)\Psi(x,t) \end{equation*}
Step 5. Inner product with $\psi_2^*(x)$. \begin{equation*} i\hbar\frac{\partial c_2(t)}{\partial t}\exp(-i\omega_2t) =2\cos(\omega t+\phi) \Bigl( c_1(t)M_{21}\exp(-i\omega_1t)+ c_2(t)M_{22}\exp(-i\omega_2t) \Bigr) \end{equation*}
where \begin{align*} M_{21}&=\int\psi_2^*(x)V(x)\psi_1(x)\,dx \\ M_{22}&=\int\psi_2^*(x)V(x)\psi_2(x)\,dx \end{align*}
Step 6. Simplify. \begin{equation*} i\hbar\frac{\partial c_2(t)}{\partial t} =2\cos(\omega t+\phi) \Bigl( c_1(t)M_{21}\exp\bigl(i(\omega_2-\omega_1)t\bigr)+2c_2(t)M_{22} \Bigr) \end{equation*}
Step 7. Define initial conditions. \begin{equation*} c_1(0)=1,\quad c_2(0)=0 \end{equation*}
For small $t$ use approximations $c_1(t)=1$ and $c_2(t)=0$. \begin{equation*} i\hbar\frac{\partial c_2(t)}{\partial t} =2\cos(\omega t+\phi)M_{21}\exp\bigl(i(\omega_2-\omega_1)t\bigr) \end{equation*}
Step 8. Solve for $c_2(t)$ by integrating. \begin{equation*} c_2(t)=\frac{2M_{21}}{i\hbar} \int_0^t\cos(\omega t'+\phi)\exp\bigl(i(\omega_2-\omega_1)t'\bigr)\,dt' \end{equation*}
Result \begin{equation*} c_2(t) =-\frac{M_{21}}{\hbar} \left( \frac{\exp\bigl(i(\omega_2-\omega_1-\omega) t\bigr)-1}{\omega_2-\omega_1-\omega} \right)\exp(-i\phi) -\frac{M_{21}}{\hbar} \left( \frac{\exp\bigl(i(\omega_2-\omega_1+\omega) t\bigr)-1}{\omega_2-\omega_1+\omega} \right)\exp(i\phi) \end{equation*}
Step 9. For $\omega$ such that $\omega\approx\omega_2-\omega_1$ the first term dominates, discard the second term. \begin{equation*} c_2(t)=-\frac{M_{21}}{\hbar} \left( \frac{\exp\bigl(i(\omega_2-\omega_1-\omega) t\bigr)-1}{\omega_2-\omega_1-\omega} \right)\exp(-i\phi) \end{equation*}
Step 10. Change to sinc function. \begin{equation*} c_2(t)=-\frac{it}{\hbar}\,M_{21} \exp\left(i\,\frac{\omega_2-\omega_1-\omega}{2}\,t-i\phi\right) \operatorname{sinc}\left(\frac{\omega_2-\omega_1-\omega}{2}\,t\right) \end{equation*}
Step 11. Transition probability. \begin{equation*} P(1\rightarrow2)=|c_2(t)|^2=\frac{t^2}{\hbar^2}\,|M_{21}|^2 \operatorname{sinc}^2\left(\frac{\omega_2-\omega_1-\omega}{2}\,t\right) \end{equation*}
Step 12. Integrate to get total transition probability. \begin{equation*} P_{tot}(1\rightarrow2)=\frac{t^2}{\hbar^2}|M_{21}|^2 \int_{E-\epsilon}^{E+\epsilon} \operatorname{sinc}^2\left(\frac{E'/\hbar-\omega}{2}\,t\right) g(E')\,dE' \end{equation*}
where $E=\hbar(\omega_2-\omega_1)$ and $g(E')$ is joint density of states.
Step 13. Use the approximation $g(E')\approx g(\hbar\omega)$. \begin{equation*} P_{tot}(1\rightarrow2)=\frac{t^2}{\hbar^2}|M_{21}|^2g(\hbar\omega) \int_{E-\epsilon}^{E+\epsilon} \operatorname{sinc}^2\left(\frac{E'/\hbar-\omega}{2}\,t\right)\,dE' \end{equation*}
Step 14. Transform the integral. \begin{align*} y&=\frac{E'/\hbar-\omega}{2}\,t \\ E'&=\frac{2\hbar y}{t}+\hbar\omega \\ dE'&=\frac{2\hbar}{t}\,dy \end{align*}
New integration limits. \begin{equation*} E\pm\epsilon \rightarrow \frac{(E\pm\epsilon)/\hbar-\omega}{2}\,t =\frac{Et}{2\hbar}-\frac{\omega t}{2} \pm\frac{\epsilon t}{2\hbar} \approx \pm\frac{\epsilon t}{2\hbar} \end{equation*}
Transformed integral. \begin{equation*} P_{tot}(1\rightarrow2)=\frac{2t}{\hbar}|M_{21}|^2g(\hbar\omega) \int_{-\epsilon t/2\hbar}^{\epsilon t/2\hbar} \operatorname{sinc}^2y\,dy \end{equation*}
Step 15. The sinc squared function is very narrow so use the approximation \begin{equation*} \int_{-\epsilon t/2\hbar}^{\epsilon t/2\hbar}\operatorname{sinc}^2 y\,dy \approx \int_{-\infty}^\infty\operatorname{sinc}^2 y\,dy=\pi \end{equation*}
Result \begin{equation*} P_{tot}(1\rightarrow2)=\frac{2\pi t}{\hbar}|M_{21}|^2g(\hbar\omega) \end{equation*}
Step 16. The transition rate is the derivative of $P_{tot}(1\rightarrow2)$. \begin{equation*} \Gamma_{1\rightarrow2} =\frac{d}{dt}P_{tot}(1\rightarrow2)=\frac{2\pi}{\hbar}|M_{21}|^2g(\hbar\omega) \end{equation*}
