When Big Jim Thompson was elected governor of Illinois in 1976 by a record 1.4 million votes, it occurred to many that the 6-foot-6 former prosecutor from the Land of Lincoln might be the next GOP presidential nominee.

"Sure," Thompson said that November, when asked about his hopes for the presidency. "I've been thinking about that since I was 11."

Well, 40 months have passed, and the crucial Illinois Republican presidential primary is only days away. And where is James R. Thompson? Still govrnor. Still popular. Still tall.

But in presidential politics he is a pygmy, a man who put aside his own ambitions, at least for new, and who just this week rationalized his own neutrality in the contest taking place in his home state, with a remarkable confession of political impotence.

"If Strom Thurmond can't carry his home county for John Connally, what can I do by endorsing someone?" Thompson asked in an interview the other day.

The decline and fall of Jim Thompson as a presidential power broker is a convenient metaphor for the frustration of Rebpublican governors as a group, in this campaign year, as in every other, for the past 20 years.

The roll call of Republicans governors who have tried to play presidential politics since 1960 is one long song of might-have-beens: Nelson A. Rockefeller, William W. Scranton, George W. Romney, Spiro T. Agnew. As would-be kingmakers or pretenders to the presidential throne, they all fell short.

This year is no different.

As the GOP battle enters its critical stage, all but two of the 19 Republican governors remain publicly neutral in the contest for their party's nomination and privately frustrated at the seemingly inexorable elimination of every alternative to Ronald Reagan, a former governor far more conservative that most of them. Reagan has the only endorsements, from Gov. Charles Thone of Nebraska and Robert F. List of Nevada.

Early on, many of the others were attracted to John B. Connally or Sen. Howard H. Baker Jr., but not strongly enough to take any real risks on their behalf.

Thompson, Ohio Governor James A. Rhodes and Texas Gov. Bill Clements all placed key allies in the Connally campaigns in their states, but none endorsed him. Tennessee Gov. Lamar Alexander endorsed his former boss, Baker, but Iowa Gov. Robert Ray waited until the Iowa caucuses were over to give his backing to baker -- a step in which he was joined by Vermont Gov. Richard B. Snelling.

Baker's campaign consultant, Douglas Bailey, said several other Republican governors were poised on the brink of endorsing Baker, "but they kept asking the scenario" by which he could win. "They didn't realize," Balley said, "that they were the scenario."

A similar Catch 22 has trapped the governors this week in regard to the possible Ford candidacy. Ford looked to them for a signal of support, and they looked the other way.

Snelling, who had tried to drum up a draft-Ford movement last fall, tried again this week to convene a meeting of Republican governors to receive a briefing on the latest draft-Ford effort, but abandoned the idea when only five governors said they would attend.

"A large number will support him if he runs," Snelling said yesterday, "but not many will recommend he run, because it's such a high-risk operation."

What explains the chronic inability of a group of men who can dominate the politics of their own states to exert their influence nationally within their party?

For many years, other governors said the problem was Rockefeller, the dominant figure in their group in terms of personality, money and political resources, but a man who polarized conservative opposition within the GOP.

Rockefeller has left the scene, yet the Republican governors appear more lost than ever. When Thompson was asked the other day why they were so ineffectual he noted that many of them are newcomers to office. This is the first nomination fight that 14 of them have seen from the perspective of the governor's office. But that is not the case with Michigan Gov. William G. Milliken, Rhodes, Ray and Otis Rowen of Indiana, who might, by themselves, provide an experienced nucleus around which the novices could rally.

Thompson also noted that "most of us get elected because of our diverse appeal to independents and Democrats, and we're not that strong in our party base."

There is some truth in that proposition but it does not negate the fact that men like Milliken, Ray, Rhodes and Bowen are the most popular and influential figures in their state parties, when they choose to exert their influence.

Snelling, who has experienced more than his share of the timidity of his colleagues makes the poing that "governors owe a lot of people in their states, not vice versa." He is alluding to the fact that many of the GOP governors have found neautrality a comfortable position when their contributors, supporters and workers are split among several presidential campaigns.

Because of that psychological barrier against a governor's leadership, Snelling said, "I never believed and do not believe now that governors have the capacity to direct events of presidential politics unless they are active early."

The chairman of the Republican Governors Association, Snelling sounds a mournful tone about the course of this year's campaign.

"I feel badly about Baker's withdrawal because, in time, people might have seen him as a thoughtful centrist," Snelling said. "I sense a rather strong feeling that Reagan ought not to be the candidate, because his presentation of the conservative position is the worst case, not the best, that can be made for it.

"A lot of people were hoping that George Bush could present the Reagan philosophy more artfully and more sensibly, but events have made him appear more strident. Now John Anderson has his chance, but I don't think it will last. I doubt very much Jerry Ford can do now what he could have done by entering last November. So, for now, I don't see any constructive course of action."

Somehow, that is what Republican governors wind up saying every four years.