Radar type application for narrow band only?
These constraints absolutely exist. There are the norm! We could only wish in our wildest dreams to use as wide as bandwidth as we like. There are many areas in a radar system that place limitations on how wide the bandwidth can be and we'll go over a few straight forward ones. Mainly we're talking about limitations due to the antenna and waveguide as well as signal processing requirements.
Antenna and Waveguide
With antennas, especially arrays, their geometries can be defined in terms of wavelengths. Given that wavelength, and therefore frequency, we can yield the antenna's directivity pattern. It's important to stress that when you typically look at an antenna pattern, it is for a single RF frequency only.
Even simple pulses have some bandwidth around the RF frequency but are usually not too large. This allows us to make the narrowband approximation that the antenna pattern is the same for frequencies near the RF frequency, which is how the pattern was defined.
With an exceptionally wideband waveform the frequencies transmitted are now farther apart and so the antenna pattern itself can change significantly intra-pulse. These affects are more extreme at lower RF frequencies where desired bandwidths are well within the order of the transmitted frequency.
These effects are undesirable because (including but not limited to):

The effective radiated power is no longer the same during the pulse.
Because the antenna pattern itself is changing during the pulse, it distorts the waveform itself and you incur SNR degradation as well as undesired phase behavior which may lead to other issues, such as errors in angle measurements.

The waveguide itself, as well as other receiver/exciter components, need to support the bandwidth of interest and thus will require the use of certain technologies and processes to support it. This is a nice way of saying that you're going to be spending a lot of money and making sacrifices in size, weight, and power (SWaP).
Signal Processing
Assuming that your RF hardware is taken care of, there are issues with data collection and front-end signal processing. High bandwidth waveforms require high sampling rates. Introducing higher sampling rates has the following overarching issues

High sampling rate analog-to-digital converters (ADCs) can be prohibitively expensive...for a good one.
Depending on the bandwidths and pulse widths desired, you may have to deal with a large number of samples. In critical applications such as a fire control radar your computing resource (usually FPGAs/CPUs) need to be able to take care of common tasks such as pulse compression, FFTs, memory copies, etc. within a very strict time schedule.

These are just a few examples. I hope it's clear that the narrowband restriction is many times the norm when designing a system. Thankfully, technology is advancing where we can get away with higher bandwidths easier than we could in the past.