The physics
Two waves from one source add their amplitudes, not their brightness. The brightness at a spot depends on the phase difference φ there.
I = I₁ + I₂ + 2√(I₁I₂) cos φ φ = 2πΔ / λ
For two equal beams of I₀ each, this becomes 4I₀cos²(φ/2).
Imax = (√I₁ + √I₂)² Imin = (√I₁ − √I₂)²
Two separate bulbs have no fixed phase. Their intensities just add, and no fringes form. Waves polarised at right angles do not interfere either.