The bandwidth of linearity and stability of a closed loop amplifier is different from the one of the open loop amplifier (op amp) used to build the closed loop system. The open loop gain varies with the frequency omega, A(omega), and depends on the internal structure of the op amp while the gain of the closed loop system is:
G(omega) = A(omega)/(1+A(omega)*beta(omega))
where beta(omega) is the frequency response of the feedback network (the two-port network you can identify as the one having its input connected to the output of the op amp, and its output connected to the feedback node of the op amp).
The A and beta needs to be evaluated in terms the Barkhausen's stability criterion which states that, in order to avoid auto oscillations, it must be that:
|Abeta| < 1 and phase(Abeta) < 45 degrees
Knowing this, one can calculate the frequency response beta(omega), given the feedback network values, and can read the A(omega) from the data sheet of the op amp.
By plotting A(omega) superposed to 1/beta(omega), one can observe what happens at the critical point where the two lines cross each other: at that critical point A = 1/beta, or conversely A*beta = 1.
The common method is to use ROC = Rate Of Closure analysis in order to study the stability around the critical point. Since the phase response of any transfer function is correlated with the slope of its frequency response, it is sufficient to look at the slope of the A(omega) and 1/beta(omega) curves where they cross each other (in a logarithmic scale plot these are the so called Bode diagrams).
The open loop A(omega) usually have a dominating pole (slope = -20dB/dec) and A(omega) crosses the frequency axis at a frequency equal to fGBP (GBP = Gain Bandwidth Product, this piece of data can be read from the op amp's data sheet), mainly due to the input capacitance of the op amp.
The feedback network of a linear closed loop amplifier is usually a voltage divider made by two resistors. This kind of network has no poles, nor zeros.
In order to have the phase margin of A*beta close to 45 degrees, it is necessary to have at least one supplementary pole in the beta(omega) diagram: this corresponds to have one zero and one pole in the 1/beta(omega) diagram, making the latter one becoming flat for high frequencies.
If we can make the the 1/beta(omega) diagram intercepting the A(omega) in a way that 1/beta(omega) is flat when A(omega) is sloping down at -20dB/dec then this situation grants the phase margin of A*beta is close to 45 degrees: the system is hence stable.
This can be done by putting a shunt capacitor Cf in parallel to the feedback resistor (the one connected to the output of the op amp in the feedback network): this solution is called the Miller's compensation. That capacitor makes the pole we need in beta(omega) in order to introduce the supplementary pole we need:
f_pole = 1/(2*piRf(Ci + Cf))
where Rf is the feedback resistor and Ci is the input capacitance of the op amp (read it from the data sheet).
The approximate triangle formed by the interception of the two curves with the horizontal axis, in logarithmic scale, gives the intercept frequency f_int to be the average of the two other vertices at the base:
log(f_int) = (log(f_pole) + log(f_GBP))/2
The latter formula can be substituted for the expression of f_pole and expanded at the first order with the Taylor series. From that expansion, inverting the formula, the proper value for the feedback capacitor can be obtained as:
Cf = (1 + sqrt(1 + 8*piRfCi*f_GBP))/(4*piRff_GBP)
A larger capacitor is still good but unnecessary: values larger than this leads to an unnecessary overcompensation.
The resulting bandwidth of the closed loop system is f_int < f_GBP, in order to grant stability. This can be read conversely: for a given desired bandwidth of the closed loop system, the bandwidth of the open loop op amp must be higher, according to what previously said. From the point of view of control theory, the control system (in this case the op amp) must always be faster than the controlled system (here being the feedback network).
This last sentences are the final answer to your question, whatever is before is the explanation of "why" it is like this.