The application of white LEDs is hindered by the low efficiency of commercial red phosphors. Here, a novel narrow-line red phosphor is produced by a terbium chain in the form of Ce³⁺–(Tb³⁺)_n–Eu³⁺ in the Na₂Y₂B₂O₇ host and is characterized with X-ray diffraction, photoluminescence (PL), PL excitation (PLE), and fluorescence lifetime, and the energy transfer (ET) processes between rare-earth ions in the host are discussed. The formation of terbium chain with a quite low content of Tb³⁺ in Na₂Y₂B₂O₇ is realized by the ET processes of Ce³⁺–Tb³⁺ and Tb³⁺–Eu³⁺, and a new concept of saturation distance is put forward as an explanation for the first time. An energy level diagram is proposed to explain the ET processes in the phosphor of Na₂Y₂B₂O₇:Ce³⁺,Tb³⁺,Eu³⁺. The emitting colour of the phosphor can be tuned from blue to green or yellow and finally to the orange–red region with increasing the content of Tb³⁺. The quantum efficiency of the phosphor with an optimized ratio of rare-earth ions, Na₂Y₂B₂O₇:0.5% Ce³⁺,60% Tb³⁺,0.5% Eu³⁺, is up to 77% under the excitation of 365 nm, which indicates that the as-synthesized phosphor is applicable to near-UV white LEDs.