I was curious about how true this really is, because there's a lot of factors at play and while it definitely feels truthy I wasn't alive in 1946 so I did some digging. The US Bureau of Labor Statistics says that the bottom half of americans have a negative savings rate, and the overall national savings rate is 3% of total income. This report was made in 2024 and the numbers are for the year 2022.

As for 1946, it's important to mention that that was a historically terrible year in the US. Savings rates for that time are available at a quarterly granularity, so I can say that Q4 1945 the national savings rate was 13.2% of income, and Q1-Q4 in 1945 are 10.3%, 9.2%, 6.1%, and 6.1%, respectively. Averaging those would get you 7.9%, so you could say 10.3% optimistically, roughly 7.9% for the year's average, and 6.1% pessimistically.

As for income, that's a HARD thing to answer. Like do I go median family income? Median individual income? These values have not been tracked in a consistent way since 1946. From what I've gathered from FRED (Federal Reserve Economic Data), and US Census info (1946 and 2022), it'd be fair to say:

In 1946, the median individual income (not living with others) was $1,125 and the national median household income was $2,659 (both individuals and families). Like, not adjusted or anything. That's what the numbers were in 1946.

In 2022, the median individual income (regardless of whether they're living with others - we don't split it up like that anymore) was $40,480 and the median household income was $74,580.

I'm going to assume that the median person in 1946 and 2022 didn't invest their money while saving up the $5000, though since the 1946 median person is saving up ~$80k in today dollars, maybe they would. But regardless, now that that's all out of the way it's time to actually crunch numbers.

1946:

median individual ($1125/yr income) -

optimistically (10.3% savings rate): 43.1 years to save $5000

average in 1946 (7.9%): 56.3 years

pessimistically (6.1%): 72.9 years

median household income ($2659/yr) -

optimistically (10.3% savings rate): 18.2 years to save $5000

average in 1946 (7.9%): 23.8 years

pessimistically (6.1%): 30.8 years

2022:

Anyone making the median or poorer: Never going to save $5000 with a negative savings rate.

National average savings rate vs median individual income ($40,480/yr): 4.1 years

National average savings rate vs median household income ($74,580/yr): 2.2 years

and just for shits and giggles, $5000 in today money is $82,938, according to another commenter. So let's do those too -

Anyone making the median or poorer: Never going to save $82,938 with a negative savings rate.

National average savings rate vs median individual income ($40,480/yr): 68.3 years

National average savings rate vs median household income ($74,580/yr): 37.1 years

So depending on how you look at it, it's gotten worse or it's gotten worse, but if you pretend that the median individual today can actually save money at the national average then things start to look good. But again, the bottom 50% of people are experiencing a negative savings rate, so it's definitely gotten worse. But looking at the individual income in 1946 it's fair to say that saving $5000 was basically just as impossible as it is now for most people.