Families of the American hostages in Tehran gathered here yesterday to discuss their frustrations and doubts with senior State Department officials as the crisis entered its fourth month.

"Really, patience wears out," said Virgil Sickmann of Krakow, Mo., whose son, Marine Sgt. Rodney V. Sickmann, is among the hostages.

About 150 relatives of the hostages spent the day at the State Department, meeting with Secretary of State Cyrus R. Vance and other officials. They were greeted by First Lady Rosalynn Carter during the afternoon.

Vance assured the families, according to a department spokesman, that the United States continues to press for the hostages' release, but he also cautioned them that he did not want to raise "false hopes" by suggesting that a breakthrough is imminent.

"You've got to wait," said Howard Albin of Little Rock, Ark., after hearing Vance. Albin's stepson, economic officer Robert O. Blucker, is a hostage. "It'll take time, won't it?" Albin asked.

Yesterday's was the third meeting here for the hostages' families since the Nov. 4 takeover of the U.S. Embassy in Tehran, according to State Department officials. The government paid for two members of each family living outside the area to travel here, officials said.

Department spokesman Hodding Carter described the meeting as designed to "share every bit of information that we have" with the families and to allow them to voice their concerns, discuss the problems they have faced and make suggestions on resolving the crisis.

Carter said some families are "so fed up after three months with what has happened that they are, in fact, asking, 'Why can't we do more?'" But he added, "There is still no short-term answer in sight."

Fifty Americans are still being held hostage in Tehran and three others remain under virtual house arrest at the Iranian Foreign Ministry, according to the State Department. Six American diplomats were smuggled out of Tehran last week by using Canadian passports and forged visas. Several of the six talked with the hostages' families yesterday.

Carter sited the escape of the six Americans as evidence the United States does not face a hopless situation in Iran.

Yet frustration was repeatedly voiced by the hostages' families yesterday. "To me, they circled around the questions that were asked of them," said Richard Hermening of suburban Milwaukee, whose son, Marine Sgt. Kevin J. Hermening, is a hostage. They have nothing concrete to offer us. [That] is really what it boils down to."

Hermening found some encouragement, nonetheless, in a briefing by a group of doctors who he said told the families that the captives would be given five days of medical and psychological checkups in West Germany after their release. "That was the most beneficial thing that we've gotten out of this," he said.