Dividend investors like to find stocks with high yields, and the best high-yield dividend stocks are ones that won't cut their payouts when times get tough. Below, we'll take a closer look at Philip Morris International (PM -0.26%), AT&T (T 0.65%), and Chevron (CVX 0.40%), and their impressive histories of consistent and growing dividend payments over time.

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Philip Morris International

Philip Morris International started out as a global tobacco giant, even though it has only been an independent company since 2008. Originally part of Altria, Philip Morris International spans the world with its Marlboro brand of cigarettes, getting most of its revenue from Europe, Latin America, and the Asia-Pacific region. Even though regulatory efforts internationally have grown at a similar pace to what the U.S. has seen, Philip Morris has fought hard to sustain its strength, and even strong currency headwinds haven't been enough to stop the tobacco giant from rewarding its shareholders.

Image: Philip Morris International.

Philip Morris International currently yields 4.3%, and it has raised its dividend payout every single year since its 2008 IPO. The most recent increase came just a month ago, and although it only boosted the quarterly payment by 2% to $1.04 per share, the company said that it has seen the strength of the U.S. dollar, which has restrained earnings growth, start to subside. With early success in marketing reduced-risk products and solid results in its traditional cigarette business, Philip Morris will have plenty of opportunities for future growth.

AT&T

The wireless-telecom space has been extremely competitive, but AT&T remains one of the largest players in the U.S. market. Even as pricing pressure has emerged from rivals' efforts to capture market share, AT&T has still found ways to differentiate itself from other carriers and hang onto most of its customer base. Moreover, the company's purchase of DirecTV helped broaden its offerings and gave it the opportunity to try to poach business from the ailing cable-television industry, and AT&T hopes that the movement of consumers away from traditional physical cable links toward internet-based video will help support its wireless business.

Image source: AT&T.

AT&T's current yield is 5.3%, and tracing its corporate roots back through the Baby Bell utility that bought the long-distance carrier and assumed its name, AT&T's streak of rising annual payouts extends back for 32 years. The company's most recent 2% increase to $0.48 per share came early in 2016, and investors shouldn't expect anything more than a similar de minimis rise in the months to come. However, with potential for business growth, AT&T investors can have confidence in the company's ability to sustain dividend levels into the future.

Chevron

The energy space has gotten hammered over the past couple of years, and falling oil prices have definitely had an impact on Chevron. Yet the oil giant's diversified holdings in upstream production, midstream distribution, and downstream refining and marketing operations has given it some advantages that pure exploration and production companies can't share. Moreover, with oil having recently climbed back toward the $50-per-barrel mark, many of those following Chevron believe that the company's cost structure and project mix give it enough flexibility to keep profits stable even at current crude price levels.

Image source: Chevron.

Chevron's yield of 4.3% isn't the highest in the industry, but it is relatively strong, and with its dividend increase earlier this week, the oil giant has a 29-year streak of increases in its total annual dividend payments to investors. The most recent 1% rise isn't a huge move for Chevron, but it does demonstrate its commitment to keep its payouts on the rise even when the business isn't facing the most favorable conditions. Chevron's strong leadership role in the oil industry should continue for years to come, and any boost in crude prices can only help prospects for its dividend going forward.

Great dividend stocks don't just have high yields. They also have the growth prospects to sustain and increase their dividends. Finding stocks like these three will help you make the most of dividend stock opportunities.