How do I make Vim do normal (Bash-like) tab completion for file names?

Apart from the suggested wildmode/wildmenu, Vim also offers the option to show all possible completions by using Ctrl + D. This might be helpful for some users that stumble across this question when searching for different autocompletion options in Vim like I did.

The closest behavior to Bash's completion should be

set wildmode=longest:full,full

With a few character typed, pressing tab once will give all the matches available in wildmenu. This is different to the answer by Michael which opens a quickfix-like window beneath the command-line.

Then you can keep typing the rest of the characters or press tab again to auto-complete with first match and circle around it.

I personally use

set wildmode=longest,list,full
set wildmenu

When you type the first tab hit, it will complete as much as possible. The second tab hit will provide a list. The third and subsequent tabs will cycle through completion options so you can complete the file without further keys.

Bash-like would be just

set wildmode=longest,list 

but the full is very handy.

If you don't want to set the wildmenu, you can always press Ctrl + L when you want to open a file. Ctrl + L will complete the filename like Bash completion.