Is it possible to combine live attenuated and non-live vaccines?
First let me point out that for the US, I'm not knowledgeable enough to speak for the rest of the world, you are correct.



There are no approved live attenuated vaccines mixed with componet/subunit/inactivated vaccines.



Background:

Before we go into why that's true, let's work a little to show that's true.  The FDA publishes a wonderful list of every approved vaccine for use in the US.  Of the 78 vaccines listed (and there is a lot of redundancy due to different manufacturers etc), only 16 are live attenuated (or about 20%).

Another thing to notice is that we have vaccines to both viruses and bacteria.   Technically, some of the vaccines are to bacterial toxins, not bacteria themselves, but for the sake of simplicity, we're going to group all bacterial component, subunit, and inactivated vaccine preparations together in one group.

First you will note that there is only one live attenuated vaccine for bacteria, Mycobacterium bovis (more on this exception later).  There are several reasons for this, but as a general rule, you don't want to put a foreign object capable of self replication inside a person.  Viruses cannot self replicate, they are obligate parasites by definition, so they don't fall under this rule.  One could, however engineer bacteria that were sufficiently deficient in replication that this might not be an issue.  Another safety concern with live attenuated bacteria is sepsis.  Again by choosing a delivery route that doesn't involve injections, you might be able to mitigate this risk.

I also want to be clear that I think live attenuated bacterial/fungal (yeast mostly) vaccines can work more broadly, and are likely to be used in innovative vaccine strategies.  

The main obstacle to a live attenuated bacteria vaccine is that the component and subunit vaccines work.  Sometimes they work even better by only supply the part of an organism that is known to provide broad neutralization (getting rid of) of the pathogen.  And if you know that you can provide protection with out the risk of putting the live organism in a person, why would you take that risk?

Quite simply you don't.  Back to M. bovis.  For the most part, humans are not vaccinated with M. bovis (BCG) to prevent M. Bovis infection (though zoonotic infections can occur).  Instead it is to prevent M. tuberculosis (MTB) infection.  Attenuation of MTB is difficult, and not complete.  Indeed we don't have a subunit or component vaccine that has worked for MTB, and an inactivated MTB has more safety concerns than BCG.  Thus we choose a related organism, M. Bovis, which naturally doesn't cause disease in humans (or rarely), but can provide some protection against TB.  Even the product insert admits that efficacy of the vaccine is in question.

The same pattern of necessity holds true for viral live attenuated vaccines.  They are used when you need a more immunogenic vaccine, and where we haven't been able to make safe and effective, inactivated, subunit, or VLP vaccines.  Thus when you need a live attenuated vaccine, you really do need it to be alive for it work.



Answering the core question:

As @MattDMo pointed out, the storage conditions are very different for live attenuated vaccines.  Not only that, but most of the preservatives used in many inactivated vaccines cause titer loss (though there is work to find alternatives). 

On the theory that a picture is worth a thousand words, I did a little experiment just for this post (and also I was cleaning out an -80).  I took a high titer stock of measles virus that has a protein tag on one of the proteins so we can easily detect it (F is Flag tagged).  Then I spiked 3 different tubes of virus in ether PBS (saline) or DTaP (an actual dose, just past expiration), and left them in the refrigerator (4oC) for 8h.  Note that Merk thinks you should toss the measles vaccine if you don't use it within 8/hours (also note their comment on preservatives).  Then I infected some cells that grow up measles, and harvested the cells in a total cell lysate.  Finally I ran equal amounts of total protein form each preparation on a gel for a western blot:



You should notice the three pairs of bands, in each case the lighter band (indicating less protein) belongs to the DTaP group.  That is to say less measles F protein (something you want to make antibodies to) came from the samples that were put into the DTaP vaccine compared to PBS.

Another serious problem is cold chain:

Again we need the virus to be "alive" when we give a person a vaccine.  This almost always means that we have to keep it frozen or at least refrigerated from the manufacturer, all the way up until moments before it goes into a person's arm.  There are many good reviews on cold chain to be found.  Quite simply, inactivated and component vaccines do not have the thermo-stability requirements that live vaccines do.  It would be a waste of energy in most cases to chill all of the vaccines which could be left at room temperature, and it would remove one of their clear advantages.

As far as the immunology is concerned, there is probably not a real limit on what the body can react to.  Indeed, as you point out, they are often administered at the same time in developed countries.  It is merely a question of how you can keep live-attenuated vaccines "alive" in the same preparation.  It is certainly possible that there is a solution to this, and vaccine stability is an active field of research.  Hopefully in 10-20 years we will be able to adjust this answer to say that such a vaccine has been made, and that it's safe enough for approval.