Early results from three state elections at the weekend provided mixed answers.
The PRI retained the Gulf coast state of Veracruz, the country's third most populous. That could be a key swing state in the next presidential elections, in the year 2000, with 4 million votes at stake. Many Mexicans believe the likely new governor, 66-year-old Miguel Aleman Velasco, son of a former president,ran merely to test his popularity and may stand down to run for president of the nation.
That would probably pit him against another former president's son, Cuauhtemoc Cardenas, of the left-wing Party of the Democratic Revolution (PRD), currently mayor of Mexico City.
But in the central state of Aguascalientes, the ruling party crashed out of power for the first time to Felipe Gonzalez, little-known until he ran for governor for the conservative National Action Party (PAN).
The PAN needed the victory to keep up momentum for the presidential race, which many commentators say the PRI could lose for the first time since it was founded. For most of those seven decades, the PRI won national, state and local elections through vote-buying or coercion. But, largely after pressure from the USA, the party was forced to open up to clean, or at least cleaner elections over the last decade.
That allowed the PAN to win six of the country's 31 states. But its momentum was slowed last month when the PRI won back one of those states, Chihuahua, proving to doubters that the long-ruling party may be ageing and infirm but it may yet have the strength to hold on to the presidency - and with it the national government - in 2000.
The PRI suffered its worst setbacks last year when, in mid-term elections, it lost its majority in the lower house for the first time and also lost the race for mayor of Mexico City - the country's second most-influential job - to PRD leader Cardenas.
The PRD ran the PRI close in Sunday's third state race, for governorship of the poverty-stricken state of Oaxaca, amid allegations of voting fraud. After early results, both parties claimed victory but most exit polls suggested PRI candidate Jose Murat was ahead.
PRD candidate Hector Sanchez disputed Murat's victory declaration and said the PRD would demand the result be declared void if fraud were proved.
Although it has not yet won a state governorship, the PRD has been chopping away at the PRI in poor southern states, where at least two guerrilla groups operate, while the strongly Catholic PAN has been spreading from its traditional power base in the north.