In the application of your bass amp and a purpose built bass cabinet, I think it is extremely unlikely you would damage a speaker in this fashion. The drivers and tweeters found in your bass cabinet are extremely durable and built to handle distortion / noise.
However, as someone who builds HiFi Stereo amps and speakers and has damaged a driver with only 3 watts, please let me share my experience:
In the HiFi, audiophile world, the drivers that comprise the “speaker system” are purpose built with very a different intention than what is found in “Pro-audio” (instrument amplification and PA systems, for example). A great example of this are HiFi tweeters of various designs: ribbons, diaphragms, silk domes, metal domes thinner than a human hair – very fragile…. Basically a whole bunch of stuff you could never use to directly amplify your bass or guitar. Some of them can handle a fair bit of power (40 or 50 watts). What they can NOT handle is a distorted signal. They are designed to provide a “flat” frequency response across a given range. Most importantly, they are designed to receive a signal that is very low distortion (this is HiFi audio, after all). Distorted signals will actually damage the mechanical materials in many of these tweeters (especially ribbons), regardless of the amount of wattage thrown at them.
For an anecdote, just look at entry level integrated stereo amplifiers – their distortion specs AT MAX OUTPUT are usually around 0.01% Total harmonic distortion. THD is much lower within the normal operating range (like 0.001% THD or less – remember, generally speaking, THD increases as an amplification device is pushed to edge of its operating limits). A clipping amplifier producing 1% THD is sending a signal that is 100 times more distorted than the max THD produced by one of these stereo amplifiers.
So how do you damage a speaker by underpowering it? If you have a low powered amp and try to make it go louder than it is designed to go, you can create a relatively high amount of THD, while still only producing a few watts. Take something like a 2A3 tube based single ended triode amp. It produces up to 3 to 3.5 watts before getting close to 1% THD (within their operating range they sound amazing, BTW). If you try and push the amp and “cheat” by increasing the gain of a pre-amp you can cause the amp to clip as much as 4% or 5%. I did this and boom – metal dome tweeter damaged (it still functioned, but crackled when played). The driver was capable of handling much more wattage. It was NOT capable of handling being fed a signal with 5% THD. You do not need to hit the driver with a square wave to damage it. These tweeters will fail long before a square wave is applied.
So, YES, in some applications, damaging a speaker by underpowering is a real thing.