1. It is not possible if the form is to be taken strictly as written; "feel sth" in the phrase given is "se sentir".
It's probably a good idea to situate the construction better first; the verb "faire" in this type of usage (i.e. that in the English sentence) must be constructed with two elements of which the second is optional according to the verb used; the construction is "faire forme verbale (qqu/qqc)" if "qqu/qqc)" is a noun.
If, in an evident way, the verb (forme verbale) can have for subject only human beings, or if it is so in a given context, then the second part (qqu), which is in plain sentences a subject to those verbs, can be either used or omitted. Here are some of those verbs:
parler, réfléchir, rire, pleurer, démanger, comprendre, frissonner, apprendre,…
vivre, cracher, souffler, souffrir,…
exemples
Le vent fait tourner la girouette. _ Le ressort fait monter le poids.
Les clowns font rire. _ Les clowns font rire les enfants. _ L'argent fait vivre. _ L'argent fait vivre ceux qui en ont.
There is sometimes the possibility of an inversion, particularly in cases of complex verbal forms.
La fin de l'été fait s'en aller les hirondelles. _ (also) La fin de l'été fait les hirondelles s'en aller.
2. When "(qqu/qqc)" is a pronoun or when the verbal form is complex, the construction is "faire (qqu/qqc) forme verbale".
exemples
Les huissiers vous font payer. _ Une seule torpille nous fera couler.
Ça fait réfléchir. _ Ça nous fait réfléchir. _ Les fortes chaleurs font languir. _Les fortes chaleurs te font languir.
After the preceding little sketch of the grammatical context, things should become simpler. The verbal form that renders "to feel sth" is "se sentir" and not "sentir", it is the pronominal form. As "(qqu/qqc)" is a pronoun (me), the construction is the second one.
"Le film me fait me sentir triste".
Quand je lui parle elle me fait me sentir comme si je ne n'était1 pas aussi intelligent qu'elle.
3. The last point mentioned is quite interesting; it is clear that both the too close morphological similarity of the two forms and the too close proximity of the pronouns wreak havoc in the average speaker's mind and that they are responsible for the mess that you can witness. There is one certitude: disregarding the strict structure of those forms and simplifying one of them makes for a blurring of the sense of discrimination, for a weakening of the habit of identification; on top of that it is certain that ambiguities too difficult to prevent will occur, if not in the case of using an adjective, as the constructions of the type "sentir triste" are meaningless, certainly in the case of using a noun. Consider the following.
- Elle était arrivée à te faire sentir du désarroi.
If the simplification is accepted, which is the assertion being made, " « te » éprouve « son propre désarroi »" or "« te » éprouve « le désarroi de « elle »»" (de « elle » ou qqu d'autre)?
This state of affairs indicates to me that sound, as simple as possible habits and habit forming are necessary; the key to such requirements does not rest with polyvalent but one-to-one correspondences.
1 correction de « comme si je ne suis pas aussi » après lecture de BBBreiz