Failed attempt to renegotiate the finance concordat (2006)
Stonewalled by the Vatican, Hungary learned in 2006 that concordat negotiations cannot be re-opened — unless the Church thinks it can get a better deal. Otherwise it's in the Vatican's interest to simply stall until a more favourable political climate ends the opposition to it. In 2010 this is what finally happened.
From the beginning there were objections to the finance concordat. Yet, despite these, the concordat was ratified by Parliament, in 1997 in “the most beautiful building of Hungary”.
More resistence was to be expected from Hungary than from Poland or Slovakia. For one thing, Hungary’s Constitution enshrined separation of church and state. (Article 60.3) For another, as a central part of the former Austro-Hungarian Empire, Hungary is both ethnically and religiously diverse, with just over half its population even claiming to be Roman Catholic. However, this very diversity enabled the Vatican finance concordat to be brought in as part of a wider religious settlement. As it was described at the time it was ratified,
Under this agreement, [and agreements with four other religious groups] the government will give $230 million in restitution to the Catholic, Lutheran, Jewish, Baptist, and Serbian Orthodox churches and synagogues. The agreement also stipulates that church-run schools are eligible for the same subsidies as public schools and that taxpayers may earmark one percent of their tax returns for a church. (According to a law passed some months ago, taxpayers could grant one percent of their income to cultural, educational, or welfare foundations. This one percent for the churches is in addition to that amount.) 
Both the non-Catholic churches and one of the coalition partners of the ruling Hungarian Socialist Party (MSZP) tried to sound the alarm. Together with the other churches, the Alliance of Free Democrats (SZDSZ) objected to the fact that the government had prepared the concordat without prior consultation with the non-Catholic establishments, violating the equality of the religious bodies. The Free Democrats even warned of what was soon to come. They objected that, according to the agreements, the state may amend taxes imposed on churches only with their consent, while they can do so unilaterally with businesses, organisations, and associations.  With this, they pinpointed one of the major objections to concordats: once ratified, they are removed from democratic control — forever.
Within ten years this danger became apparent to all. In April of 2006 a coalition once more led by the Hungarian Socialist Party (MSZP) was re-elected on a platform promising economic “reform without austerity.” However, this was not feasible: by the time of the election the budget deficit had ballooned to over 10% of annual GDP (the largest in the 25-nation European Union) and needed to be brought down towards 3%, in the hope of eventually qualifying for adoption of the Euro. Hungary's current account deficit, the broadest measure of an economy's trade performance, was not much smaller. The stock market in Budapest plunged and a prominent American economist said that "Hungary is an accident waiting to happen."  Faced with this fiscal impasse, the government naturally thought of the huge payments flowing to Rome.
In late July, therefore, István Hiller, who was both leader of the socialist party and Minister of Education and Culture, established contact with the Holy See, carefully observing Vatican diplomatic protocol. Then he flew to Rome to negotiate a review of the finance concordat. After all, the first concordat had talked of establishing "mutual relations of friendship" and now it came to the test. However, the Hungarian Minister found himself stonewalled: no one at the Vatican was available to enter into serious negotiations with him! 
After this setback, the Minister returned home and in early September 2006, his Government announced the European Monetary Union Convergence Plan. This was a series of austerity measures to try to reduce the deficit in order to eventually qualify for the Euro. By the end of the month the hardship this imposed had led to rioting in the streets of Budapest.
Then in the April 2010 elections the Vatican saw its chance. A conservative and Catholic coalition (Fidesz and KDNP) of gained the two-thirds parliamentary majority required to change the constitution. The new constitution enshrines key Catholic doctrines outlawing abortion and permitting discrimination against gays.  Needless to say, the Vatican no longer faces any opposition from Hungary over the onerous financial concordat.
1. “Constitution Watch”, East European Constitutional Review, [New York University School of Law] , Volume 6 Number 4, Fall 1997.
2. “Constitution Watch”.
3. “Hungary faces enormous economic hurdles”, USA Today, 2006-06-23
4. “Education minister returns empty handed from Vatican trip”, Hungary Around the Clock, 1 August 2006.
5. “Hungary’s new constitution: Family friendly, hostile to gays”, Euractive, 28 March 2011. http://www.euractiv.com/future-eu/hungarys-new-constitution-family-friendly-hostile-gays-news-503455