High-altitude clouds and hazes are integral to understanding exoplanet observations, and are proposed to explain observed featureless transit spectra. However, it is difficult to make inferences from these data because of the need to disentangle effects of gas absorption from haze extinction. Here, we turn to the quintessential hazy world, Titan, to clarify how high-altitude hazes influence transit spectra. We use solar occultation observations of Titan's atmosphere from the Visual and Infrared Mapping Spectrometer aboard National Aeronautics and Space Administration's (NASA) Cassini spacecraft to generate transit spectra. Data span 0.88-5 μm at a resolution of 12-18 nm, with uncertainties typically smaller than 1%. Our approach exploits symmetry between occultations and transits, producing transit radius spectra that inherently include the effects of haze multiple scattering, refraction, and gas absorption. We use a simple model of haze extinction to explore how Titan's haze affects its transit spectrum. Our spectra show strong methane-absorption features, and weaker features due to other gases. Most importantly, the data demonstrate that high-altitude hazes can severely limit the atmospheric depths probed by transit spectra, bounding observations to pressures smaller than 0.1-10 mbar, depending on wavelength. Unlike the usual assumption made when modeling and interpreting transit observations of potentially hazy worlds, the slope set by haze in our spectra is not flat, and creates a variation in transit height whose magnitude is comparable to those from the strongest gaseous-absorption features. These findings have important consequences for interpreting future exoplanet observations, including those from NASA's James Webb Space Telescope.