You might assume that if you save nicely for retirement, you won't end up so reliant on Social Security once your career comes to an end. But the reality is that you don't know exactly how well your savings will hold up during your retirement years.

You might wrap up your career with a $1 million nest egg. But if you withdraw from your savings too aggressively and/or the market sorely underperforms, you could end up with a lot less income than initially expected. And in that situation, a higher monthly Social Security benefit could truly come to the rescue.

That's why it's so important to claim Social Security at the right time. And while you might think that signing up for benefits at age 65 is a smart choice, it's a decision you might really end up regretting after the fact.

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Even a smaller hit to your monthly benefits could be detrimental

The earliest age to claim Social Security is 62. But you're not entitled to your full monthly benefit based on your personal income history until full retirement age (FRA) arrives. FRA is either 66, 67, or 66 and a specific number of months, depending on the year you were born.

Now you may be thinking it's a good idea to claim Social Security at age 65 because that's when Medicare eligibility begins. And to be clear, enrolling in both programs at the same time could make it easier and more convenient to pay your Medicare Part B costs.

Medicare Part A, which covers hospital care, is generally free for those who enroll (meaning free in terms of premiums -- there are still out-of-pocket costs you might incur when you use your Part A coverage, like deductibles for hospital stays). But Medicare Part B, which covers outpatient care, charges a monthly premium that can vary depending on different factors, such as your income.

Seniors who are enrolled in Social Security and Medicare at the same time have their Part B premiums deducted from their benefits automatically. But you don't have to be receiving Social Security to enroll in Medicare.

If you're not yet signed up for benefits, you'll simply pay your Part B costs directly. And if you make the decision to sign up for Social Security at the same time as Medicare for the convenience factor, you might end up with a financial crunch on your hands.

Claiming Social Security at age 65 could mean taking a hit of up to 13.34% on your benefits (that full 13.34% reduction will apply if your FRA is 67). So if you'd normally get $2,000 a month from Social Security at age 67, filing at 65 will mean getting $1,733 a month instead.

That hit might seem insignificant at first -- especially if you have a healthy nest egg. But imagine you end up spending down your savings sooner than anticipated. At that point, you might really come to miss that extra $3,200 a year in retirement income.

Waiting until FRA often pays

Claiming Social Security at age 65 will mean taking a modest hit on your benefits -- not nearly the same hit you'd be looking at as filing for benefits at age 62. In spite of that, waiting until FRA to sign up for Social Security could really be a more advantageous move, especially if you're not desperate to start receiving those benefits around your 65th birthday.

It's one thing to claim Social Security at age 65 if your career has wrapped up and you need the income. But if that's not the case, waiting a short while longer could really help you in the long run.

And if you're not convinced, think of how the stock market has performed in the past year. Even with a nest egg that's only modestly invested in stocks, an extended downturn could have a huge impact. So the more money you can lock in from Social Security, the less financial stress you might have as a retiree.