Is a 16f628a + pbp capable of multithreading? Is it capable of letting a led flash while proceeding another task to symbolize that it is busy?
You know and I know, based on previous performance, that nobody, anywhere, not today, not tomorrow, not ever, will be able to answer your questions to your satisfaction nor will you heed advice given to you by others...

But just in case....

No, PICs are not capable of multi-threading. None of them are. Period. Case closed. PICs cannot complete the execution of more than one single cycle instruction at any given time. Read it, learn it, live...
PC's up until about 10 years ago were never capable of multi-threading or multi-tasking as we think of the process today. They all used interrupts (either hardware or software) to switch tasks and/or memory space quickly enough that it seemed like they were running more than one program at a time.

It is very easy to, as you put it, fade and LED in and out, while running another program in a main loop, using interrupts. If I had a video of it, I would show you 20 RGB LEDs connected to a PIC18F8720, all doing something different, fading in/out, changing color, and so on and so on, while at the same time, displaying various debugging messages on an LCD, reading a 5 button keypad, reading a light 'program' from an SD card, communicating over a CP2103 USB link, and so on. All done with interrupts...not multi-tasking/-threading. All because it was being done so fast that it appeared to all be done at once. Slow down the oscillator speed to something very, very, slow and a person would be able to see (thru an ICD) the program executing different loops of code at different times.

That should be clear enough...but you'll probably refute every statement made above...and I'll probably 'un-refute' you at every turn....