Near-field propagation is often unavoidable at terahertz (THz) frequencies due to the large apertures needed for sufficient array gain, yet near-field operation complicates practical system design, especially under user mobility. This paper asks whether a mobile THz link can remain broadband, achieve the desired high rates and coverage, while operating exclusively in the radiative far field. To answer this question, we develop a proof-by-contradiction feasibility framework that jointly enforces (i) a far-field requirement based on the Fraunhofer distance and (ii) a reliability requirement specified by a target SNR at the worst-case link distance. We derive closed-form upper bounds on the far-field-feasible bandwidth for stationary and mobile links. We further incorporate practical misalignment through several UE rotation and mobility scenarios. Numerical results show that stationary THz links can remain far-field-only with physically realizable apertures while supporting extremely large bandwidths, whereas practical mobile THz systems cannot. In practically relevant mobile THz access settings, the far-field-feasible bandwidth becomes a severe limiting factor: achieving tens-of-GHz targets would require unrealistically high UE transmit power. A cross-band comparison further shows that far-field-only operation is largely attainable at sub-6~GHz and, to a significant extent, at mmWave for moderate bandwidths, while near-field-aware designs become essential for mobile THz access.