This mostly happens when the maximum coverage has not been attained, e.g., in dilute systems and/or when the adsorption capacity of adsorbent is too high. As an attempt to obtain distinguishable results, one can conduct an experiment using extended range of adsorbate concentration towards higher levels.

In any case, drawing mechanistic conclusions alone from isotherm fitting should be avoided. Isotherms results should be better discussed in conjunction with results obtained from studies on kinetics and thermodynamics. Having information about the adsorbent's characteristics, especially the morphology and the surface chemistry, also provide useful supporting evidence during interpreting the adsorption behaviors.

The first question is whether there was a proper check for adequate descritpion of the experimental data by these two models. Use of correlation coefficients is generally not sufficient. Sum of squares of deviations, for calculated minus experimental values, should be examined for statistical significance, with accounting for experimental scatter. One good test is "lack-of-fit" (may be found in the texbooks). Both models may be found good but yet not describing the data adequately. Both models may be found describing data adequately, or only one. In the case if both models seem to describe the data adequately, i..e, deviations of th the fitted data from the experimental values show no trend and may be just assigned to random scatter, I would say, it is not possible to choose between a model demonstrating a saturation (Langmuir) and the model typically interpretated as associated with exponential distribution of the sites acording to the energy. In other words, we cannot choose between two conceptual points, an one type of sites or variety...Both hypotheses fit the experimental data. So, what we could learn yet from the data? One certain conclusion is that sorption isotherm is non-linear, and there is a gradual saturation of sorption site(s). To distinguish between the models, one may either extend solute concentration, as suggested by Mr. Mushariv Khan, or vary, for example, temperature. If the isotherm measured at another temperature fitted by Langmuir model shows a different maximal capacity, the Langmuir model is conceptually not suitable (since the capacity is not supposed to change with temperature). In principle, there is one more, "delicate" analysis: Langmuir model tends to linear isotherm, when reducing solute concentrations whereas Freundlich model never tends to linear isotherms. So, by carefully examining the data distribution along the lower solute range, this could be also revealed.

A very useful and informative review, it might help you to well understand both of the mentioned models.

These videos can answer your questions: https://www.youtube.com/channel/UC5MiS3nMoji95OZ7uE1V8qw/playlists?view=50&sort=dd&shelf_id=6