Up to which dilution can one assume that 1 mg/l = 1 ppm?
"Parts per …" are dimensionless entities: to derive $\pu{ppm}$ from mass concentration $γ$ (e.g. $\pu{1 mg L-1})$ one uses density of water $\rho(\ce{H2O}) = \pu{1E6 mg L-1}$ for normalization:

$$\frac{γ}{ρ(\ce{H2O})} = \frac{\pu{1 mg L-1}}{\pu{1E6 mg L-1}} = \pu{1E-6} = \pu{1 ppm}$$

Once the solution is too concentrated, you obviously are not allowed to use $ρ(\ce{H2O})$ and have to use the real concentration of the solution.
However, this occurs at the concentration levels way higher than those to be expressed with $\pu{ppm}$ anyway: that's why "parts per …" are so ubiquitously used in biochemistry. Higher molar masses, lower molar concentrations, primarily aqueous media: no need to account for the error stemming from the difference in the densities of pure water and the solution.

Note that "parts per something" units are deprecated and are not recommended for use, see e.g. Is 1 ppb equal to 1 μg/kg?.