This 15-digit accuracy fails for numbers that outside the valid range. For example, the number 1e500 is too large and cannot be directly represented using standard 64-bit floating-point numbers. Similarly, 1e-500 is too small and it can only be represented as zero.
The range of 64-bit floating-point number might be defined as going from 4.94e-324 to 1.8e308 and -1.8e308 to -4.94e-324, together with exactly 0. However, this range includes subnormal numbers where the relative accuracy can be small. For example, the number 5.00000000000000e-324 is best represented as 4.94065645841247e-324, meaning that we have zero-digit accuracy.
For the 15-digit accuracy rule to work, you might remain in the normal range, e.g., from 2.225e−308 to 1.8e308 and -1.8e308 to -2.225e−308. There are other good reasons to remain in the normal range, such as poor performance and low accuracy in the subnormal range.
To summarize, standard floating-point numbers have excellent accuracy (at least 15 digits) as long you remain in their normal range which is between 2.225e−308 to 1.8e308 for positive numbers.