Yes, there is a connection between losing one phonemic property and gaining another. Most approaches to phonology conceptualize words as having double lives: on the one hand, they are made of a particular sound sequence which you have to pronounce correctly; on the other hand, the sounds in sequences are only recognized as discrete parts because they contrast with other sounds.
This property of phonology was termed double articulation by the French phonologist Martinet. One also speaks of form vs. substance: form being properties of speech sounds which they have by virtue of being in contrast with other sounds, or undergoing meaning-preserving alternations involving other sounds, and substance being the physical phonetic details of their pronunciation. (Quite confusingly, one also hears of function vs. form, where form coincides with substance from the other, roughly equivalent, dichotomy.)
These two factors frequently come into play in the evolution of vowel systems. To characterize it approximately: Consider a set system of 14 vowels, with 7 basic vowel qualities and two lengths. For such a system, the pertinent formal properties are that each vowel is contrastive with every other, and that the vowels may also be divided into pairs (e.g., e and e:) on the basis of meaning-preserving alternations. The fact that the pairs are differentiated by length and not by quality instead relates to the substance.
When a language undergoges sound change such that the phonetic substance is altered, but the formal relations between the sounds are preserved, it is usually referred to as transphonologization. Such a process is quite common historically, because languages do have a tendency of conservatism in form, if not in substance.
For further reading and numerous examples, consider a 2008 paper by the phonologist Larry Hyman and references included therein. For further reading treating English phonology more specifically, see various studies by Roger Lass, esp. English Phonology and Phonological Theory, and Old English Phonology.
EDIT: I waited ten hours for other answers to appear, then presented my own findings. I do not pretend that mine is the only possible answer, and would like to hear what others have to say in their own answers.
The English word playa is pronounced /ˈplaɪ.ə/ in English with two syllables, with the dot there representing a syllabic boundary.
English doesn’t really have triphthongs because no phonemic sequence of three vowel sounds occupies the same syllable in English. As with a diphthong, all components of a triphthong must occur in the same syllable.
English phonetician John Wells, author of the Longman Pronunciation Dictionary, doesn’t believe that vowel sequences traditionally considered “triphthongs” like /aɪə/ actually are such. In his blog posting “John Wells’s phonetic blog: triphthongs anyone?”, he takes issue with the traditional British idea that those are actually triphthongs:
Some authors describe the English vowel system as including not only diphthongs but also triphthongs. Peter Roach (English Phonetics and Phonology,
4th ed., CUP 2009, p. 18-19) puts it like this:
The most complex English sounds of the vowel type are the triphthongs. They can be rather difficult to pronounce, and very difficult to recognise. A triphthong is a glide from one vowel to another and then to a third, all produced rapidly and without interruption.
He lists the triphthongs eɪə, aɪə, ɔɪə, əʊə, aʊə (later giving the example words layer, player; liar, fire; loyal, royal; lower, mower; power, hour) and continues [...] I find this account unsatisfactory. If the əʊə of slower is a “triphthong”, it is difficult to see any reason why the əʊɪ of going is not one too. If liar has a triphthong, surely trying must have one. [...] Similarly, I would argue that part of the definition of a true triphthong must be that it constitutes a single V unit, making with any associated consonants just a single syllable.
Given that, do we have triphthongs in English? I claim that generally, at the phonetic level, we don’t. I treat the items we are discussing as basically sequences of a strong vowel plus a weak vowel. (By ‘strong vowel’ I mean one that is stressable and the potential input to a weakening rule; by ‘weak vowel’ I mean one that is the potential output of a weakening rule. Diphthongs such as aɪ are included under the heading ‘strong vowel’.)
So according to Professor Wells, English doesn’t generally have triphthongs at the phonetic level, which he explains in more detail at the referenced article.
Therefore English playa has no triphthong at all, but rather two syllables where the stressed one has a ‘strong’ vowel — here a diphthong — and the second and unstressed syllable has a separate ‘weak vowel’ that is not part of the first syllable.
As Janus mentions in a comment, there may be English dialects in which certain phonemes have allophones that can produce a tautosyllabic triphthong. However, this is not phonemic, only phonetic, because there are no minimal pairs to prove an actual phonemic distinction.
Best Answer
Traditional RP has 12 monophthongs: six short vowels—kit, put, dress, strut, trap, lot— five long vowels—fleece, goose, nurse, thought, start and the schwa—banana.
According to this blog post, Modern RP has up to three more monophthongs—square, near, cure. These are essentially long versions of the short vowels dress, kit, put. However, it also says that in modern RP, the vowels in fleece and goose are turning into diphthongs. But I suspect that a lot of speakers don't follow either of these systems exactly. So this gives 15 vowels that might be pronounced as monophtongs in RP (some of these are long and short versions of the same vowel), and 10 vowels that generally are.
And the number of diphthongs isn't constant in American English, either—the vowel of ride is often a monophthong in Southern American English.