Is it possible to have a lip enhancement filler and 'tear trough' under eye fillers in one day?

Yes, you can have these treatments on the same day. Keep in mind, a combination approach will give the most improvement for under the eyes. Fillers, lasers, and microneedling/PRP can improve under the eyes. My patients try to get non surgical treatments to avoid surgery. We use PRP and stem cells there often. I suggest consulting with an expert to develop a treatment plan best suited for your short and long term goals. Best, Dr. Emer

There is no problem doing fillers at two or more areas on same day. Its done commonly to give a uniform enhancement. 

There would be no problem having a filler placed into the lips and into the tear trough areas on the same day.  This combination is commonly performed.    

This is an excellent question. These procedures can definitely be done the same day. I recommend preparing a few days in advance with Arnica and discontinuing supplements that may cause bruising as these are areas that are very vascular. Ice packs and ample cooling time will help make these procedures very comfortable and minimize swelling during and after treatment. A consultation prior to treatment will give you all the steps needed to prepare for these procedures. 

Short answer is YES! Remember that it is quite common to having bruising (usually mild) with lower eyelid/tear trough filler and swelling with lip filler. You may want to have a few days after where you don't have much going on just in case.

Consider a full face evaluation with a physician who you can trust to help you make an informed decision on how to approach your concerns in a conservative way that also meets your goals. Always remember that for any treatment you should always be offered FDA-approved options. Expertise and technic is necessary for rapid recovery and excellent results. An evaluation though is necessary beforehand for proper recommendations to make sure you are not doing too much at once.
Best, 

It's absolutely possible to have both in one day. Fillers in both of these areas can create a dramatic change and tear troughs especially are considered an advanced technique...so just make sure you find a board-certified dermatologist or plastic surgeon who has experience with dermal fillers and other injectable treatments.
I hope this helps!
Dr. Gilly