Basically, no.

I've answered a couple previous questions on this; last month about the development and purpose behind the literacy tests and one a few months earlier on how they were implemented in practice.

In short, whether or not you were educated or wealthy made very little difference when the overall goal was to disenfranchise anyone who was targeted by it, black or occasionally white. There were rare instances like Abernathy where someone figured out a way around the test to get registered (at substantial risk to their health if and when they went to the polls), but I've not seen any evidence of Blacks being able to run for even local government positions after the implementation of Jim Crow laws became widespread in the South in earnest after Plessy.

Outside a tiny handful of regions where the Republican party was mostly "Lily-White" (versus the "Black-and-Tan" coalition that had been formed during Reconstruction), the only government positions in the South that continued to have any Republicans whatsoever occupying them were some postmaster jobs if you were politically connected enough. This was an offshoot of the quarter of Republican delegates who came from the South for Presidential conventions up through the early 1900s, when roughly half of that group were Black. These were considered 'the rotten boroughs', and their votes were either controlled sometimes by patronage appointments but often by outright bribery; it depended on how contested the nomination was. Southern convention representation continued at a slightly lower rate (~15%) through the 1960s, although Blacks began to be slowly purged from Southern party leadership through the 1910s and 1920s with at least the tacit acceptance of party leaders, including Taft, Harding, and Coolidge. This picked up considerably with the active encouragement of Hoover to the point where several states had no Blacks whatsoever in party leadership from the end of his term until the 1970s, although in others some Black political bosses held on despite this.

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Some could, not just because they were educated but because they were connected to powerful institutions.

The City of Tuskegee, Alabama and surrounding Macon County show the limits of what was possible before the Voting Rights Act of 1965. Macon County lay in Alabama's Black Belt of majority-black farming counties but differed from them because Tuskegee contained institutions, most notably Tuskegee Institute (now Tuskegee University) that fostered the development of an independent black middle class whose members met even the rigged requirements for voting. I would argue that the key to this different trajectory is these institutions themselves and not simply the notable individuals they attracted such as Booker T. Washington and, more relevant to the period under discussion, Charles G. Gomillion (1900-95), who taught sociology at Tuskegee Institute from 1928 to 1971 and spearheaded efforts to organize black citizens as voters and protect their political rights.

Tuskegee Institute was founded in 1881 as Reconstruction was waning and black citizens were losing political rights. The Institute had a longstanding policy of only employing black people. It received much of its funding from northern philanthropists, so it was not dependent on local white people or the state legislature: in fact, the Institute brought a lot of money into the community for which white people competed. Its faculty and administrators were comparatively well-paid and could be housed on the campus (rather than rent from white landlords).

In 1923, the federal government added another black-empowering institution when the Veterans Administration established a hospital in Tuskegee to serve black veterans. Critically, the VA decided to employ an all-black medical staff at this hospital. This added another group of educated, well-paid black employees who were likewise not dependent on local white people. In addition, during WWII the U.S. Army Air Force opened Tuskegee Army Air Force base to train black aviators, staffing it primarily with black servicemembers and civilians, so this temporarily added yet another group of educated, economically-independent black citizens. The large black middle class and the institutions they ran had customer relationships with local white businessmen.

The populations of both Macon County and Tuskegee were about eighty percent black, so white political control depended upon suppressing the black vote, primarily by imposing obstacles to registration. On paper these barriers were formidable. Under Alabama's post-Reconstruction constitution adopted in 1901, to register to vote you either had to be both literate and employed or possess property worth more than $300--though some registrars treated this "or" as though it were an "and." In addition, you had to pay a poll tax. Whether a citizen had met the requirements was left to the judgment of a three-member board of registrars chosen by the state (not county) government, though they had to be locals.

This was effective at disenfranchising nearly all black (and many white) citizens in most of the state, but in Macon County there was a nucleus of black citizens who met the requirements: they were not just literate but well-educated, had secure jobs, often possessed the requisite property, and could afford the poll tax.

Macon County's board of registrars had also adopted to practice of requiring that any black registrant find two white voters to vouch for him as a suitable voter. Here is where black economic power facilitated by the Institute and VA Hospital was especially important. White businessmen could hardly refuse to vouch for their most valued black customers, such as the Institute and VA Hospital's administrators who both drew comfortable salaries themselves and had substantial control over institutional spending. Crossing the color line to let a valued customer vote did not seem like a big transgression, especially when there were just a few dozen black voters out of perhaps 2,000 in the county. When Gomillion came to Tuskegee in 1928, he learned there were just 32 black voters, all of whom were in the administration of the Institute and VA Hospital.

Gomillion's own successful quest for registration is instructive. In 1934 he decided to try to register. He was literate, employed, and had money and property. The only obstacle was finding white voters to vouch for him. The white dry goods merchant from whom his wife made regular purchases was willing. The second voucher was a white building contractor, whom Gomillion induced in 1939 to vouch as a prerequisite to a lucrative contract to build the Gomillion family a new home. Cannily, Gomillion waited until the board of registrars had added him to the voter list before signing the contract, which gave the contractor every incentive to lean on the board rather than allow it throw up more obstacles to registering the assertive scholar.

A couple years later in 1941, another Institute professor obtained a white merchant's promise to vouch for him after he made an expensive installment purchase. That merchant repeatedly failed to show at the courthouse (though his other white voucher, a haberdasher, followed through), so eventually Gomillion (who was now leading the local civil rights struggle) threatened to sue the board over the "two white vouchers" requirement and other irregularities (the "or/and" issue mentioned above). The somewhat chastised board decided to allow two established Institute administrators--who were black--to vouch for the professor and be done with the issue. Going forward, he board continued to impose the vouching requirement but sometimes accepted long-standing black vouchers.

Gomillion also championed another lawsuit to win the vote for another individual only to have the board of registers claim in the midst of the litigation, almost certainly fraudulently, that he had been registered all along and the paperwork had just been misplaced.

So the white local officials were at times overcome by the passion, qualification, and economic power of middle-class black citizens backed by those independent institutions. As one put it in 1939, "We hold it down all we can, but we got this Veterans' Hospital and the Institute here. They're educated, and they've got property, and there's not much you can do."

Meanwhile, Gomillion and others were ensuring the growing number of black voters acted as a bloc, which given the small total number of voters in the city and county meant candidates did well to court their votes. Gomillion reformed the Tuskegee Men's Club as the Tuskegee Civic Association and gave it a political purpose.