One of the biggest disagreements about conversion is kabbalot ol mitzvot, the acceptance of the yoke of the commandments.
In case you don't remember, yoke mean burden, not to be confused with a yolk. Fun fact: the "stock" from "being put in the stocks" is a human yoke. Criminals and slaves were put in stocks. Sounds exciting to take on the mitzvot now, right? Don't worry, the fun doesn't end there. Marriage was also traditionally analogized to the yoke.
The Gemara says that if a conversion candidate says, "I will keep your Torah, except this one thing," the rabbi shouldn't convert him or her. Of course, rabbis have found much to disagree about even in such a straight-forward-sounding sentence, especially when combined with other discussions in the Talmud like Rabbi Hillel's conversion of a man who agreed only to accept the Written Torah but not the Oral Torah. Rabbis argue whether Hillel actually performed the conversion while he still believed that or only after he came to trust Hillel's teachings that the Oral Torah exists and is binding on all Jews. This case seems to be the major wrench in this entire debate.
This halacha is most often applied to the case of a woman dating or married to a kohen. A female convert can't marry a kohen (women can't be kohanim, so it doesn't matter for men). So if a woman says, "I want to marry Joe Bob the Kohen here, I want to be converted," then the rabbi cannot convert her because she is showing the rabbis that she doesn't intend to keep the prohibition of a convert not marrying a kohen. And this case gives the rabbis the right to infer the intent to not keep a mitzvah based on the actions of the candidate. As the rabbinic rulings evolve over time, there are cases where the rabbis choose to avoid the inference. Rabbis can also choose to not ask the question, as Rabbi David Hoffman said in the case below.
At least one, and probably more, cases involve women being converted to marry a kohen. The one I remember is an exceptionally unusual case from the late 1800s/early 1900s: a gentile woman married a kohen secularly. Their baby died, and she mistakenly assumed the baby was a Jew because his father was a Jew. She became very upset and distressed. This distress was created by the sadness that she and her baby were not of the same religion, and the rabbi was afraid that she might go mad. So the rabbis decided to let her convert. I think this case has very little application for anyone else. And if you want to try to make it applicable by going mad, I think they're more likely to institutionalize and medicate you long before they convert you. To be fair, the rabbi did not allow her to have a Jewish marriage with her kohen husband. He ruled that it would be "better" for the couple to live together without a Jewish marriage. That last part is included solely for your amusement. It's not relevant to our discussion.
Going back to the "I want to marry so-and-so" hypothetical above, lack of kabalat ol mitzvot is generally the reason why a person cannot be converted while pursuing a relationship with a non-observant Jew. A house divided cannot stand, and many converts have lost their partner for being "too Jewish." Without both partners committed to orthodox observance, the convert is not going to be able to observe all of the mitzvot since some involve both sides of a marriage, plus the issues of kashrut and Shabbat (and maybe even sabotage of them by the partner!). While there used to be significant debate about whether such candidates should be accepted anyway (up until just a couple of decades ago?), that debate is mostly closed today, especially since the Israeli Rabbinate has pushed the diaspora states to take a machmir stance on this issue. Performing those kinds of conversions could bring significant disrepute on a rabbi's other conversions today.
Now I don't want to scare you, but a failure to have kabalat ol mitzvot at the time of conversion is one of the things that makes a conversion void, as though it never existed. And that's where the trouble lies: someone could argue that you didn't have the intention to fulfill all the mitzvot at the time of conversion. The allegations could be based on true events or made-up ones depending on how unlucky you are, but you're already in an "elite" crowd if you ever have this issue. But getting back to the issue, it's almost impossible to prove what was in someone's mind as they stood in the mikvah. Arguing there was no kabalat ol mitzvot can be very difficult unless the candidate immediately violated the mitzvot and continued to do so. This is the subject of the argument over Russian conversions in Israel.
(This same non-observance is a major reason why non-orthodox conversions are not accepted as being legitimate conversions. Because of the alleged failure of even the rabbis - who are responsible for teaching the convert how to be "Jewish" - to observe all the mitzvot as a general rule, there is a presumption that the converts cannot have the halachically required kabalat ol mitzvot at the time of immersion in the mikvah. That is why a liberal conversion that has a beit din, bris, and mikvah could be ruled un-kosher, but there is also the argument that non-observant Jews on the beit din - including the rabbi - make the beit din an invalid beit din, and thus, they cannot approve a conversion.)
Background on the Russian conversion issue: Soviet Jewry was a big deal. Google it if you have no idea what I'm talking about. When the iron curtain fell, Jews and descendants of Jews poured into Israel to begin a new life. These people had been labeled as Jews by the Soviet government and suffered for being Jews. But many were not halachicly Jewish, which doesn't matter for the purpose of making aliyah, but eventually these immigrants want to get married, etc. There is no civil marriage in Israel, only religious marriage, and the Rabbinate are the arbitrars of who is Jewish enough for a Jewish marriage. Having always affiliated as a Jew but being secular, many of these Russians are having issues resolving their status because they are not willing to take on kabalat ol mitzvot - and I'm glad they're generally honest enough to say so! But they still want the conversion, and some rabbis are turning to these older responsa that provide alternate ways of defining kabalat ol mitzvot or trying to avoid the issue altogether. Sephardim are generally more lenient in these matters, but the entire thing has been a mess. I don't foresee it getting resolved anytime soon.
But I guess you want "the answer" to the subject of this post: Can you still convert if you refuse to do X, Y, and Z? I'm no rabbi, but I would say no. However, whether those topics come up in conversation so that the rabbis are aware of your refusal is a different matter. However, in today's orthodox conversions, it is highly unlikely you will convert if you don't appear to be accepting everything and living a fully Torah-observant life. The totally secular will not convert unless there were some serious considerations in favor of the conversion and a rabbi willing to stick his neck out for it (they exist). As a practical matter, when you stand in the mikvah (and maybe beforehand too), you will be asked whether you agree to certain things. Acceptance of both the Oral and Written Law and the rabbinic interpretation and enactments will be one of those things. So I hope you're prepared to answer honestly.