Is there a rule, on how many Patron Saints a Catholic can have?

I believe you're referring to the fact that a person baptized needs a "Christian Name" and that name would be their protection throughout their life.

The sacrament of Baptism is conferred "in the name of the Father and of the Son and of the Holy Spirit." In Baptism, the Lord's name sanctifies man, and the Christian receives his name in the Church. This can be the name of a saint, that is, of a disciple who has lived a life of exemplary fidelity to the Lord. The patron saint provides a model of charity; we are assured of his intercession. The "baptismal name" can also express a Christian mystery or Christian virtue. "Parents, sponsors, and the pastor are to see that a name is not given which is foreign to Christian sentiment

CCC 2158

When your saint's "name day" comes around, it's a bit like your own spiritual birthday, etc.. It's a good an pious practice hat should probably be done more. This is usually your own name, unless your parents named you something that's not particularly Christian and then you get a new name at Baptism.

The code of Catholic Church law issued in 1918 instructed pastors to see to that each person baptized was given a “Christian” name. If the parents of a child to be christened insisted on an “un-Christian” name, the baptizing priest was instructed to add a Christian name on his own initiative and to enter both names in the baptismal register.

The revised code of Church law (1983) is less restrictive. It simply forbids the giving of a baptismal name that is “alien to Christian sentiment.” Most likely, however, this broader rule will not have much influence on the established custom of preferring the names of saints. This is good, for it is better for us to have a heavenly patron saint whom we can admire and turn to. - Baptismal Saints

However, I don't think it's universally recognized that you can only have one patron, especially since that patron was probably chosen for you by your parents and most likely doesn't have a whole heck of a lot to do with your vocation or particular needs. St. Theresa of Avila, who was a bit of a rule breaker - so maybe this isn't a good example, took St. Joseph as her patron to help her in her work.

and I took for my patron and lord the glorious St. Joseph, and recommended myself earnestly to him.

Life of Teresa of Jesus

She believed that he could help her with everything.

I took the name Alphonsus, because I liked Frank Zappa and that was the closest I could get to St. Alfonso (of pancake breakfast fame). But he did guide me quickly back into the Church when I bought a big book from Ligouri press on a whim that turned out to be a Catechism (which I don't recall having seen in Catechism class).

Digressions aside, I believe you must have several saints that you are devoted to in order to survive as a Catholic and I don't believe that one will be saddened by your veneration of another.

St. Teresa heard from the next highest authority too, telling us in her autobiography, “She (our Blessed Mother) said that I pleased her very much by being devout to the glorious St. Joseph.”

National Catholic Register

Lastly, and this is just a practice that may be applicable to my diocese, given the way we structure baptisms and confirmations, but when confirmation candidates in High School prepare for the sacrament, they're tasked with finding a saint to go and make their confirmation patron. It's never incumbent upon them to choose their Baptismal name, although that might be a lapsed practice that uninformed catechists, like myself, would be better off correcting.

How many patron saints may we have at any one time even if temporarily?

There seems to be no fixed rule on this subject.

Before going on allow me to relate a little story. I once asked the retired Bishop James Timlin of Scranton, Pennnslyvania which patron saint he took for his Name's Day, whether it was St. James the Less or St. James the Greater. He told us: "I take both of them for my Name's Day!

What is a Patron Saint?

Patron Saint are Saints:

•who by designation of the sovereign pontiff or by popular tradition are venerated as favouring by their intercession certain interests, countries or localities, e.g., Saint Joseph, patron of the Universal Church, Saint Francis Xavier, patron of Catholic Missions, Saint Camillus of Lellis, patron of hospitals

•after whom churches, parishes, or other institutions are named

•after whom persons are named in Baptism or Confirmation

Our permanent patron saints are usually in the area of two and if we add our Guardian Angel that takes it to three.

Although not a patron saint in the strict sense of the word, they are nevertheless assign to us by God to help us in all our ways.

Saint Thomas Aquinas holds that, “from the very moment of his birth man has an angel guardian appointed to him” (Summa Theologica, I, 113, 5). Even more so, Saint Anselm states that at the very moment of the union of soul and body God appoints an angel to watch over him/her. This would mean that during pregnancy a woman would be surrounded by two guardian angels. They watch over us from the very beginning and it is up to us to allow them to fulfill their duties for the rest of our lives. - 5 Facts You Need to Know About Your Guardian Angel

As for truly permanent patron saints they are usually given to us at our baptism and our confirmation (in the Latin Rite). They are known as our "Name's Day Saints". Eastern Rite Churches confirm their infants at the moment of baptism.

Some religious take a religious name as a patron saint and thus have an addition spiritual helper for them in heaven.

A name day is a tradition in some countries in Europe, Latin America, Catholic and Orthodox Christian countries in general. It consists of celebrating a day of the year that is associated with one's given name. The celebration is similar to a birthday.

The custom originated with the Christian calendar of saints: believers named after a saint would celebrate that saint's feast day, or in the Eastern Orthodox tradition, the day of a saint's death. Name days have greater resonance in the Catholic and Orthodox parts of Europe—Protestant churches showing less veneration of saints. In many countries, however, name-day celebrations no longer have connection to explicitly Christian traditions. - Name day (Wikipedia)

Other patron saints may be ether permanent or temporary according to our personal situations, such as occupations or medical conditions.

For example if one is blind, there are at lest 19 patron saints to chose from. One could have a devotion to any one of them or have a devotion to each and every on of them.

Nowhere have I read there we must be limited to just one patron saint per condition or occupation, nor are we obliged to keep a special devotion to a particular patron saint (Name's Day Saint). The Church leaves us with a great amount of freedom in this area.