If you can completely block out any frequency from the input that would make it oscillate, then the overall system will also not oscillate. However as LvW pointed out, stability of a system is not defined that way. Also, constructing such an input is impossible to do.
- For example, any non-periodic signal is bound to have spectral content over higher frequencies (can be limited, but not eliminated). So just switching on the input will generate spectral content with higher frequencies causing the oscillations to start. Any component that changes even slightly (eg. temperature, vibration, etc.) will also result in higher frequency spectral content.
- Then there is noise, which is also typically white (present over all frequencies). Even a simulator can introduce quantization noise, or computational errors, also leading to high-frequency content. Noise is usually inherently present and can never be eliminated.
So there is no real-life situation where band-limiting the input signal helps. However, in theory it certainly is possible. The negative feedback loop situates itself in an unstable singular state, where even the smallest perturbation will cause oscillation.
On a side note: What you're describing is sometimes the case in simulation. When designing oscillators (where there is no input, or the input is \$0\$ having no spectral content at all), the simulator often calculates such an unstable singular state with such high precision that it won't start oscillating (or that it starts very slowly). Ie. you then usually need to apply some perturbation to it yourself to make it start. Usually the quantization noise is enough to get it started (very slowly) though.