Thank you for a very informative response!
I agree with the comment about "misinformation" in this post, but not surprised.
First, the comment that since you fell short of 300DPI (capability of the printer????), is incorrect. Depending on the capability of the printer, I get excellent prints using as low as 225DPI, and do often print at 250 to 275 DPI. Beyond a certain point, extra DPI is simply wasted ink that takes longer to dry or absorb into the surface of the paper. In fact, most printers in home situations will print to 360DPI, but ink is surely wasted.
Test your printer on various print settings of DPI from 225 to 300. Optimally 300 is usually the best, but may be wasting ink, where 275 DPI may be excellent as well, with a savings on ink. You did NOT mess up your print by falling 12 DPI short of someone's optimal.
Secondly, I checked the post rather thoroughly, and see NO mention of the amount of pixels in the image you printed from. 72DPI will print like crap normally, if you throw it out to the printer at that res. That is for monitor viewing, and web site use only.
HOW many megapixels were in your image, and remember that if you crop you lose megapixels.
The math for printing is complex. If there are not enough pixels to really create a 16X20 print (you will need 28.8 megapixels in the image to print best sharpness at 300DPI), you compromise the print of the image. This is why I often print at 250DPI (at which point you would need only 20 megapixels to reach the dimension at 250 DPI)
Since I don't know what size pixelwise your original image was after any cropping, I will surmise you did not have 20 to 28 megapixels.
In that case, your print function of your editing program if you force the issue by changing the print dimensions x DPI, your editing program will be forced to interpolate additional pixels to reach those dimensions and DPI.
Now regular editing software is just not up to this task in a way that interpolates truly usable pixels by the printer.
It is best to use software that is designed to do this "upping of resolution" with a final result of good print quality. One such was mentioned in a prior post as a plug in to the edit program you may be using. There are also free standing programs that will provide quality UPRESSING, as it is called.
Without understanding the process of increasing the resolution of files to print quality prints, starting with too few pixels in the first place, it will be difficult to get a handle on why the quality of a printed image is so variable.
It's based on first, mathematically understanding when you go beyond the optimal print size due to the megapixels you have to work with. Beyond that, you increase the dimension of the print with an upward gain in resolution (more pixels) and a grasp of what DPI is best for your printer. That varies with the subject content. So proofing is one way of doing that.
NOTE: Pardon me, but I used 16x20 on my math of printing examples, instead of the OP 16X24. my mistake, but idea the same.
The best book I have run across on the math of printing is a book called "The Magic of Digital Printing" by Derek Doeffinger. The ISBN number is ISBN - 10: 1-57990-689-3, or ISBN - 13: 978 - 1 - 57990 - 689 - 4
Very good informative soft cover 8X11 w 160 pages