Are there any conditions at all that allow a married Catholic couple to use contraceptives?

I was thinking for many hours to provide some kind of more positive answer that still answers the direct original question, without verging into apologetic.

The add-on question about sterilization pointed the way.

Quite apart from any theological points, there's a basic medical rationale behind all of this which says that it's basically unsound to deliberately alter a healthy, major, functioning organ or system of the body. This applies even to the level of contraception, if it ain't broke don't fix it and there's a built-in mechanism (cycles and awareness aka science) for running with the grain that predates the technological "fix".

The Catholic understanding of the body is like this, and the radical nature of its application to relationships (marriage and family) implies in all those "tough" examples that probably it's society that ought to change. A bit like how the mainstream concept of disability is now that society ought remove barriers to participants, it's not them that have missed the boat. It's the boat that needs a rebuild.

No, there are not—at least not for contraceptive purposes.

The Catholic Church has this to say about (artificial) contraception:

Similarly excluded [from consideration as a moral act] is any action which either before, at the moment of, or after sexual intercourse, is specifically intended to prevent procreation—whether as an end or as a means.

Neither is it valid to argue, as a justification for sexual intercourse which is deliberately contraceptive, that a lesser evil is to be preferred to a greater one, or that such intercourse would merge with procreative acts of past and future to form a single entity, and so be qualified by exactly the same moral goodness as these. Though it is true that sometimes it is lawful to tolerate a lesser moral evil in order to avoid a greater evil or in order to promote a greater good," it is never lawful, even for the gravest reasons, to do evil that good may come of it—in other words, to intend directly something which of its very nature contradicts the moral order, and which must therefore be judged unworthy of man, even though the intention is to protect or promote the welfare of an individual, of a family or of society in general. Consequently, it is a serious error to think that a whole married life of otherwise normal relations can justify sexual intercourse which is deliberately contraceptive and so intrinsically wrong.

(Humanae Vitae, Encyclical letter of Blessed Pope Paul VI, July 25, 1968, section 14; emphasis added)

Pope Paul reasons thus to this conclusion:

This particular doctrine, often expounded by the magisterium of the Church, is based on the inseparable connection, established by God, which man on his own initiative may not break, between the unitive significance and the procreative significance which are both inherent to the marriage act.

The reason is that the fundamental nature of the marriage act, while uniting husband and wife in the closest intimacy, also renders them capable of generating new life—and this as a result of laws written into the actual nature of man and of woman. And if each of these essential qualities, the unitive and the procreative, is preserved, the use of marriage fully retains its sense of true mutual love and its ordination to the supreme responsibility of parenthood to which man is called. We believe that our contemporaries are particularly capable of seeing that this teaching is in harmony with human reason.

(Humanae Vitae, section 12)

Pope St. John Paul II, in his apostolic exhortation Familiaris Consortio of 1981, said something similar:

When couples, by means of recourse to contraception, separate these two meanings that God the Creator has inscribed in the being of man and woman and in the dynamism of their sexual communion, they act as "arbiters" of the divine plan and they "manipulate" and degrade human sexuality—and with it themselves and their married partner—by altering its value of "total" self-giving. Thus the innate language that expresses the total reciprocal self-giving of husband and wife is overlaid, through contraception, by an objectively contradictory language, namely, that of not giving oneself totally to the other. This leads not only to a positive refusal to be open to life but also to a falsification of the inner truth of conjugal love, which is called upon to give itself in personal totality.

When, instead, by means of recourse to periods of infertility, the couple respect the inseparable connection between the unitive and procreative meanings of human sexuality, they are acting as "ministers" of God's plan and they "benefit from" their sexuality according to the original dynamism of "total" selfgiving, without manipulation or alteration.

(Familiaris Consortio, section 32)

That is, because contraception involves one or the other of the couple changing the way the physiology of sex works, the Church sees it as the couple manipulating their sexuality and in a sense withholding something about themselves from the other. This is a fundamental dishonesty which has no place in the vocation of marriage; in addition, it is an act of pride, in that it represents the couple deciding that they, not God, will control whether life will come of their love.

As hinted above, the Church recognizes that there may be times when a couple wishes to ensure that sex will not result in having a child. In such cases, there are morally acceptable approaches (including what are often called fertility awareness methods) which allow one, without interfering with the body's normal functioning, to discover the times when the woman is most and least likely to conceive, and to refrain from sexual activity during the "most likely" times.

But there are no times when artificial contraception methods are morally acceptable.

It is, of course, the case that hormonal methods of contraception have other physical effects on the woman's body, and these are taken advantage of when the medications are prescribed to deal with other gynecological conditions. To take a medication for the purpose of curing a disease is never immoral, and Pope Paul VI addressed this possibility in Humanae Vitae:

On the other hand, the Church does not consider at all illicit the use of those therapeutic means necessary to cure bodily diseases, even if a foreseeable impediment to procreation should result therefrom—provided such impediment is not directly intended for any motive whatsoever.

However, especially in light of the potential health complications, one might consider asking the gynecologist to prescribe such medications only as a last resort.

It's too bad that so many Catholics appear to not know, or perhaps to not care, what the actual teaching of the Church is on this subject.